CORNELL UNIVERSITY LIBRARIES Mathematics Library White Hall CORNELL UNIVERSITY LIBRARY 3 1924 059 316 103 DATE DUE rETTB \m 1 1 1 GAYLORD PRINTCOIN U.S.A. The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924059316103 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1991. BOUGHT WITH TH^J^^COM^ FROM THE' •■•-*» SAGE ENDOWMENT FUND THE GIFT OF Denrg M. Sage 1891 .11' INTRODUCTION INFINITESIMAL ANALYSIS FUNCTIONS OF ONE REAL VARIABLE BY OSWALD VEBLEN Preceptor in Mathematics, Frznceton University AND N. J. LENNES Instructor in Malhemaiia m the WejidelL PktUipa High Schoolt Chicago FIRST EDITION FIRST THOUSAND NEW YORK JOHN WILEY & SONS London: CHAPMAN & HALL, Limited 1907 Copyrigdt, 1907 BI OSWALD VEBLEN awd N. J. LENNES ROBERT DRCHHOHD, PRINTER, NEW TORE PREFACE. A COURSE dealing with the fundamental theorems of infini- tesimal calculus in a rigorous manner is now recognized as an essential part of the training of a mathematician. It appears in the curriculum of nearly every university, and is taken by students as "Advanced Calculus " in their last collegiate year, or as part of "Theory of Functions " in the first year of graduate work. This little volume is designed as a convenient reference book for such courses; the examples which may be considered necessary being supplied from other sources. The book may also be used as a basis for a rather short theoretical course ou real functions, such as is now given from time to time in some of our universities. The general aim has been to obtain rigor of logic with a minimum of elaborate machinery. It is hoped that the system- atic use of the Heine-Borel theorem has helped materially toward this end, since by means of this theorem it is possible to avoid almost entirely the sequential division or " pinching " process so common in discussions of this kind. The definition of a Umit by means of the notion "value approached" has simplified the proofs of theorems, such as those giving necessary and sufficient conditions for the existence of _ limits, and ia general has largely decreased the number of e's and d's. The theory of limits is developed for multiple-valued functions, which gives certain advantages in the treatment of the definite integral. In each chapter the more abstract subjects ?ind those which can be omitted on a first reading are placed in the concluding IV PREFACE. sections. The last chapter of the book is more advanced in character than the other chapters and is intended as an intro- duction to the study of a special subject. The index at the end of the book contains references to the pages where technical terms are first defined. When this work was undertaken there was no convenient source in English containing a rigorous and systematic treat- ment of the body of theorems usually included in even an ele- mentary course on real functions, and it was necessary to refer to the French and German treatises. Since then one treatise, at least, has appeared in English on the Theory of Functions of Real Variables. Nevertheless it is hoped that the present volume, on account of its conciseness, will supply a real want. The authors are much indebted to Professor E. H. Moore of the University of Chicago for many helpful criticisms and suggestions; to Mr. E. B. Morrow of Princeton University for reading the manuscript and helping prepare the cuts; and to Professor G. A. Bliss of Princeton, who has suggested several desirable changes while reading the proof-sheets. CONTENTS. CHAPTER I. The System of Real Numbers. PAoa { 1. Rational and Irrational Numbers 1 J 2. Axiom of Continuity 3 § 3. Addition and Multiplication of Irrationals 7 J 4. General Remarks on the Number System 11 § 5. Axioms for the Real Number System 13 § 6. The Number e 15 § 7. Algebraic and Transcendental Numbers 18 § 8. The Transcendence of e 19 § 9. The Transcendence of ;: 25 CHAPTER II. Sets of Points and of Segments. § 1. Correspondence of Numbers and Points 30 § 2. Segments and Intervals. Theorem of Borel 32 § 3. Limit Points. Theorem of Weierstrass ; 38 § 4. A Second Proof of Theorem 15 42 CHAPTER III. PoNcnoNS IN General. Spbclal Cases of Functions. § 1. Definition of a Function 44 5 2. Bounded Functions 47 { 3. Monotonic Functions. Inverse Functions 49 { 4. Rational, Exponential, and Logarithmic Functions 63 CHAPTER IV. Theory of Limits. § 1. Definitions. Limits of Monotonic Functions 60 § 2. The Existence of Limits 65 vi CONTENTS. PAGS § 3. Application to Infinite Series 70 § 4. Infinitesimals. Computation of Limits 74 § 5. Further Theorems on Limits 81 § 6. Bounds of Indetermination. Oscillation .,.,,., 83 CHAPTER V. CoNTiNuoDs Functions. § 1. Continuity at a Point 87 § 2. Continuity of a Function on an Interval 88 J 3. Functions Continuous on an Everywhere Dense Set 94 § 4. The Exponential Function 97 CHAPTER VI. Infinitesimals and Infinites. § 1. The Order of a Function at a Point 101 § 2. The Limit of a Quotient 105 § 3. Indeterminate Forms 108 § 4. Rank of Infinitesimals and Infinites 114 CHAPTER VII. Derivatives and Differentials. { 1. Definition and Illustration of Derivatives 117 § 2. Formulas of Differentiation itq § 3. Differential Notations 128 § 4. Mean-value Theorems joq % 5. Taylor's Series , , . I 6. Indeterminate Forms , og J 7. General Theorems on Derivatives 144 CHAPTER VIII. Definite Integrals. { 1. Definition of the Definite Integral 151 § 2. Integrability of Functions , c^ 5 3. Computation of Definite Integrals . cq § 4. Elementary Properties of Definite Integrab IfM { 5. The Definite Integral as a Function of the Limits of Integration 171 § 6. Integration by Parts and by Substitution j^ • § 7. General Conditions for Integrability ,-_ CONTENTS. vii CHAPTER IX. Improper Definite Integrals. PAGE 5 1. The Improper Definite Integral on a Finite Interval 191 § 2. The Definite Integral on an Infinite Interval 201 § 3. Properties of the Simple Improper Definite Integral 205 § 4. A More General Improper Definite Integral 210 § 5. Special Theorems on the Criteria of the Existence of the Improper Definite Integral on a Finite Interval 218 } 6. Special Theorems on the Criteria of the Existence of the Improper Definite Integral on an Infinite Interval 223 INFINITESIMAL ANALYSIS. CHAPTER I. THE SYSTEM OF REAL NUMBERS. § 1. Rational and Irrational Numbers. - The real number system may be classified as follows: (1) All integral numbers, both positive and negative, in- cluding zero. (2) All numbers — , where m and n are integers (n^^O). (3) Numbers not included in either of the above classes, such as V2 and w.f Numbers of classes (1) and (2) are called rational or com- mensurable numbers, while the numbers of class (3) are called irrational or incommensurable numbers. As an illustration of an irrational number consider the square root of 2. One ordinarily says that \/2 is 1.4 +, or tit i8 clear that there is no number — such that — 7=2, for if — r = 2. n n' ' n' ' then m'=2n', where m' and 2n' are integral numbers, and 2n' is the square of the integral number m. Since in the square of an integral number every prime factor occurs an even number of times, the factor 2 must occur an even number of times both in n' and 2n', which is impossible because of the theorem that an integral number has only one set of prime factors. 2 INFINITESIMAL ANALYSIS. 1.41 + , or 1.414 + , etc. The exact meaning of these statements is expressed by the following inequalities: t (1.4)2 <2< (1.5)2, (1.41)2 <2< (1.42)2, (1.414)2 <2< (1.415)2, etc. Moreover, by the foot-note above no terminating decimal is equal to the square root of 2. Hence Horner's Method, or the usual algorithm for extracting the square root, leads to an infinite sequence of rational numbers which may be denoted by Oi, a2, as, . . . , a„, . . . (where ai=1.4, 02 = 1.41, etc.), and which has the property that for every positive integral value of n a„<o„+i, a„2<2< ("- + 1^)' Suppose, now, that there is a least number o greater than every o„. We easily see that if the ordinary laws of arith- metic as to equality and inequality and addition, subtraction, and multiplication hold for a and a^, then a^ is the rational nimiber 2. For if a2<2, let 2-o2 = e, whence 2=a? + e. II n were so taken that jjc<-r, we should have from the last in- 2 £ e <0„2 + 4g + g<02+£. equality { so that we should have both 2=a^ + e and 2<a2 + e. On the t a < 6 signifies that a is less than b. a>b signifies that a is greater than 6. t This involves the assumption that for eveiy nvimber, e, however small there is a positive integrer n such that ttt-K t- This is of course obvious 10** o when e is a rational number. If c is an irrational number, however, the statement will have a definite meaning only after the irrational number has been fully defined. THE SYSTEM OF REAL NUMBERS. 3 other hand, if o^ > 2, let a^ - 2 = e' or 2 + e' =a. Taking n such that r-r- < — , we should have lU" 5 (a« + J^)' < (fln^) + e' < 2 + e' < a; and since Cn + iT^ is greater than a* for all values of k, this would contradict the hypothesis that a is the least number greater than every number of the sequence a\, 02, 03, . . . We also see without difficulty that o is the only nimiber such that «2 = 2. § 2. Axiom of Continuity. The essential step in passing from ordinary raticwial num- bers to the number corresponding to the sjonbol \/2 is thus made to depend upon an assumption of the existence of a number o bearing the unique relation just described to the sequence a^, a2, a„, . . . In order to state this hypothesis in general form we introduce the following definitions: Definition. — ^The notation [x] denotes a set,'f any element of which is denoted by x alone, with or without an index or subscript. A set of numbers [x] is said to have an upjxr bound, M, if there exists a number M such that there is no number of the set greater than M. This may be denoted by M^[x]. A set of numbers [x] is said to have a lower bound, m, if there exists a number m such that no number of the set is less than m. This we denote by m^[x}. Following are examples of sets of numbers: (1) 1, 2, 3. (2) 2, 4, 6, . . . , 2i, . . . (3) 1/2, 1/22, 1/23, . . . , i/2», . . . (4) All rational numbers less than 1. (5) All rational numbers whose squares are less than 2. t Synonyms of set are class, aggregate, collection, assemblage, etc. 4 INFINITESIMAL ANALYSIS. Of the first set 1, or any smaller number, is a lower bound;, and 3, or any larger number, is an upper bound. The second set has no upper bound, but 2, or any smaller nimiber, is a lower bound. The nimiber 3 is the least upper boimd of the first set, that is, the smallest niraiber which is an upper bound. The least upper and the greatest lower bounds of a set of num- bers [x] are called by some writers the upper and lower limits respectively. We shall denote them by B[x] and 5[x] respect- ively. By what precedes, the set (5) would have no least upper bound xmless V2 were coimted as a number. We now state our hjrpothesis of continuity in the following form: Axiom K. If a set [r] of rational numbers having an upper bound has no rational least upper bound, then there exists one and only one number B[r] such that (a) B[r] > /, where / is any number of [r] or any rational rmmr- ber less than some number of [r]. (6) B[r]<r", where r" is anyrational upper bound of [rj.f Definition. — ^The number ^r\ of axiom K is called the least upper bound of [r], and as it cannot be a rational number it is called an irrational nimiber. The set of aU rational and irrational numbers so defined is called the cordinuxms real num- ber system. It is also called the linear continuum. The set of all real numbers between any two real numbers is likewise called a linear continuum. Theorem i. // two sets of rational numbers [r] and [s], having upper bounds, are such that no r is greater than every s and no s greater than every r, then B[r] and B[s] are the same; that is, in symbols. Proof. — ^If B[r] is rational, it is evident, and if ^r] is irra- tional, it is a consequence of Axiom K that B[r]>s', t This axiom implies that the new (irrational) numbers have relations of order with all the rational numbers, but does not explicitly state rela- tions of order among the irrational numbers themselves. Cf. Theorem 2. THE SYSTEM OF REAL NUMBERS. 5 where s' is any rational number not an upper bound of [s]. Moreover, if s" is rational and greater than every s, it is greater than every r. Hence where s" is any rational upper bound of [s]. Then, by the definition of B[s], B[r]=m Definition. — If a number x (in particular an irrational ■number) is the least upper bound of a set of rational numbers [r], then the set [r] is said to determine the number x. Corollary 1. The irrational numbers i and i' determined by the two sets [r] and [/] are equal if and only if there is no num- ber in either set greater than every number in the other set. Corollary 2. Every irrational number is determined by some set of rational numbers. Definition. — If i and i' are two irrational numbers deter- mined respectively by sets of rational numbers [r] and [/] and if some number of [r] is greater than every niunber of [/], then i>i' and i'<i. From these definitions and the order relations among the rational numbers we prove the following theorem: Theorem 2. If a and b are any two distinct real numbers, then a<b orb<a; if a<b, then not b<a; if a<b and b<c,then a<c. Proof.— Let o, 6, c all be irrational and let [x], [y], [2] be sets of rational numbers determining a, b, c. In the two sets [x] and [y] there is either a number in one set greater than every number of the other or there is not. If there is no number in either set greater than every number in the other, then, by Theorem 1, a=b. If there is a number in [x] greater than every number in [y], then no number in [y] is greater than •every number in [x]. Hence the first part of the theorem is 6 INFINITESIMAL ANALYSIS. proved, that is, either a=b or a<b orb <a, and if one of these^. then neither of the other two. If a number yi of [y] is greater than every number of [x], and a number zi of [z] is greater than every number of [y], then zi is greater than every num- ber of [x]. Therefore if o<6 and b<c, then a<c. We leave to the reader the proof in case one or two of the numbers a, b, and c are rational. Lemma. — If [r] is a set of rational numbers determining an irrational number, then there is no number ri of the set [r] which is greater than every other number of the set. This is an immediate consequence of axiom K. Theorem 3. If a and b are any two distinct numbers, then there exists a rational number c such that a<c and c<b, or b<e and c<a. Proof. — Suppose a<b. When a and b are both rational —^ is a nimiber of the required type. If a is rational and b irrational, then the theorem follows from the lemma and Corol- lary 2, page 5. If a and b are both irrational, it follows from Corollary 1, page 5. 11 p, is irrational and b rational, then there are rational numbers less than b and greater than every number of the set [x] which determines a, since otherwise b would be the smallest rational nimiber which is an upper bound of [x], whereas by definition there is no least upper bound of [x] in the set of rational numbers. Corollary. A rational number r is the least upper bound of the set of all numbers which are less than r, as well as of the set of all rational numbers less than r. Theorem 4. Every set of numbers [x] which has an upper bound, has a least upper bound. Proof. — Let [r] be the set of all rational mmabers such that no number of the set [r] is greater than every number of the set [x]. Then B[r] is an upper boimd of [x], since if there were a number Xi of [x] greater than B[r], then, by Theorem 3, there would be a rational nmnber less than Xi and greater than B[r], which would be contrary to the definition of [r] and B[r]. THE SYSTEM OF REAL NUMBERS. 7 Further, B[r]js the least upper bound of [x], smce if a number N less than B[r] were an upper bound of [x], then by Theorem 3 there would be rational numbers greater than N and less than B[r], which again is contrary to the definition of [r]. Theorem 5. Every set [x] of numbers which has a lower bound has a greatest lower bound. Proof. — ^The proof may be made by considering the least upper bound of the set [y] of all numbers, such that every num- ber of [y] is less than every niunber of [x]. The details are left to the reader. Theorem 6. If all numbers are divided into two sets [x] and [y] such that x<y for every x and y of [x] and [y], then there is a greatest x or a least y, but not both. Proof. — The proof is left to the reader. The proofs of the above theorems are very simple, but ex- perience has shown that not only the beginner in this kind of reasoning but even the expert mathematician is likely to make mistakes. The beginner is advised to write out for himself every detail which is omitted from the text. Theorem 4 is a form of the continuity axiom due to Weier- strass, and 6 is the so-called Dedekind Cut Axiom. Each of Theorems 4, 5, and 6 expresses the continuity of the real num- ber system. § 3. Addition and Multiplication of Irrationals. It now remains to show how to perform the operations of addition, subtraction, multipUcation, and division on these numbers. A definition of addition of irrational numbers is sug- gested by the following theorem: "If a and b are rational num- bers and [x] is the set of all rational niunbers less than a, and [y] the set of all rational numbers less than b, then [x+y] is the set of all rational nvmibers less than a+b." The proof of this theorem is left to the reader. Definition. — ^If a and b are not both rational and [x] is the set of aU rationals less than a and [y] the set of all rationals less 8 INFINITESIMAL ANALYSIS. than b, then a+b is the least upper bound of [x + y], and is called the sum of o and b. It is clear that if b is rational, [x+b] is the same set as [x+y]; for a given x+b is equal to x' + (6-(x'-x))=a;'+2/', where x' is any rational number such that x<x'<a; and conversely, any x+y is equal to (i - 6 + j/) + 6 = i" + 6. It is also clear that a+b=b+a, since [x+y] is the same set as [y+x]. Likewise {a+b) +c=a + {b+c), since [{x+y) +z] is the same as [x + {y+z)]. Furthermore, in case b<a, c=B[x'—y'], where a<x' <b and a<y'<b, is such that b + c=a, and in case b<a, c=^x' —y'] is such that b + c==a; c is denoted by o— & and called the differ- ence between a and b. The negative of a, or — o, is simply 0— a. We leave the reader to verify that if a>0, then a+b>b, and that if o<0, then a+b<b for irrational numbers as well as for rational*'. The theorems just proved justify the usual method of add- ing infinite decimals. For example : ;: is the least upper bound of decimals hke 3.1415, 3.14159, etc. Therefore n:4-2 is the least upper bound of such nimibers as 5.1415, 5.14159, etc. Also e is the least upper bound of 2.7182818, etc. Therefore r+e is the least upper bound of 5, 5.8, 5.85, 5.859, etc. The definition of multiphcation is suggested by the follow- ing theorem, the proof of which is also left to the reader. Let a and b be rational numbers not zero and let [x] be the set of all rational numbers between and a, and [y] be the set of aU rationals between and b. Then if a > 0, 6 > 0, it follows that ab = B[xy] ; a<0, b<0, " " " ab = B[xy]; a<0, 5>0, " " " a6 = 5[a;j/]; a>0, 6<0, " " " ab = B[.xyl Definition. — If a and 6 are not both rational and [a;] is the set of all rational numbers between and o, and [y] the set of all rationals between and 6,_then if a>0, b>0, ab means B[xy]; if a<0, 6<0, ab means ^xy]■, if o<0, 6>0, ab means B[xy]; if a>0, 6<0, ab means B^xj/]. If a or 6 is zero, then ab=0. THE SYSTEM OF REAL NUMBERS. 9 It is proved, just as in the case of addition, that ab=ha, that a{bc) = idb)c, that if a is rational [ay] is the same set as [xy], that if a>0, b>0, ab>0. Likewise the quotient -r- is defined as a number c such that ac=b, and it is proved that in case a>0, 6>0, then c=b\ —/\, where [y'] is the set of all rationals greater than b. Similarly for the other cases. More- over, the same sort of reasoning as before justifies the usual method of multiplying non-terminated decimals. To complete the rules of operation we have to prove what is known as the distributive law, namely, that o(6+c)=a6+ac. To prove this we consider several cases according as a, b, and c are positive or negative. We shall give in detail only the case where all the numbers are positive, leaving the other cases to be proved by the reader. In the first place we easily see that for positive numbers e and /, if [t] is the set of all the rationals between and e, and [T] the set of all rationals less than e, while [u] and [U] are the corresponding sets for /, then e+f=B[T+U]=B[t+u]. Hence if [x] is the set of all rationals between and a, [y] be- tween and b, [z] between and c, b + c=B[y+z] and hence a{b+c) = B[x{y+z)]. On the other hand ab=B[xy], (ic=^^xz], and therefore db+ac= ^{xy+xz}]. But since the distributive law is true for rationals, x(y+z)=xy+xz. Hence E[x{y+z)]=B[ixy+xz)] and hence aib + c)'=ab+ac. We have now proved that the system of rational, and irra- tional nimabers is not only continuous, but also is such that we may perform with these nimibers all the operations of arith- metic. We have indicated the method, and the reader may 10 INFINITESIMAL ANALYSIS. prove in detail that every rational number may be represented by a terminated decimal, aaO*+Oifc_ilO*-i + . . .+ao+~-+ •■■ +^ = afcfflt_i . . . aoffi_ia_2 . . . ffl_n, or by a circulating decimal, O'kP'k-l • • • ffloffl-lffl-2 ■ . ■ d-i . . . d- id-i ■ . ■ a_j . . . , where i and j are any positive integers such that i<]'; whereas every irrational nimiber may be represented by a non-repeating infinite decimal, aicO-ic-i ■ ■ ■ aoa_ia_2 . . . «_„ . . . The operations of raising to a power or extracting a root on irrational numbers wiU be considered in a later chapter (see page 53). An example of elementary reasoning with the sym- bol jB[x] is to be found on pages 17 and 18. For the present we need only that x", where n is an integer, means the number obtained by multipl3dng x by itself n times. It should be observed that the essential parts of the defini- tions and arguments of this section are based on the assumption of continuity which was made at the outset. A clear under- standing of the irrational number and its relations to the rational number was first reached during the latter half of the last century, and then only after protracted study and much discussion. We have sketched only in brief outline the usual treatment, since it is beUeved that the importance and diffi- culty of a full discussion of such subjects wUl appear more clearly after reading the following chapters. Among the good discussions of the irrational number in the English language are : H. P. Manning, Irrational Numbers and their Representation hy Sequences and Series, Wiley & Sons, New York; H. B. Fine, College Algebra, Part I, Ginn & Co., Boston- THE SYSTEM OF REAL NUMBERS. 11 Dedekind, Essays on the Theory of Number (translated from the German), Open Court Pub. Co., Chicago; J. Piehpont, Theory of Functions of Real Variables, Chapters I and II, Ginn & Co., Boston. § 4. General Remarks on the Number System. Various modes of treatment of the problem of the number system as a whole are possible. Perhaps the most elegant is the following : Assume the existence and defining properties of the positive integers by means of a set of postulates or axioms. From these postulates it is not possible to argue that if p and V q are prime there exists a number a such that ap=q or a = -, i.e., in the field of positive integers the operation of division is not always possible. The set of all pairs of integers jm, n), if \mk, nk\ {k being an integer) is regarded as the same as {Tn,n\, form an example of a set of objects which can be added, subtracted, and multiplied according to the laws holding for positive integers, provided addition, subtraction, and multipli- cation are defined by the equations,t {m,n\®{p,q\ = {mp,Tiq} {m, n!01p, 5! = {mq+np, nq]. The operations with the subset of pairs {m,l} are exactly the same as the operations with the integers. This example shows that no contradiction will be introduced by adding a further axiom to the effect that besides the integers there are numbers, called fractions, such that in the extended system division is possible. Such an axiom is added and the order relations among the fractions are defined as follows: p m .. -<— if pn<qm. an t The details needed to show that these integer pairs satisfy the alge- braic laws of operation are to be found in Chapter 1, pages 5-12, of Pier- pont's Theory of Real Functions. Pierpont's exposition differs from that indicated above, in that he says that the integer pairs actuaUy are the frac- tions. 12 INFINITESIMAL ANALYSIS. By an analogous example t the possibility of negative num- bers is shown and an axiom assuming their existence is justi- fied. This completes the rational nimiber system and brings the discussion to the point where this book begins. Our Axiom K, which completes the real number system, assuming that every bounded set has a least upper bound, should, as in the previous cases, be accompanied by an exam- ple to show that no tontradiction with previous axioms is intro- duced by Axiom K. Such an example is the set of all lower segments, a lower segment, S, being defined as any Jsounded set of rational numbers such that if a; is a number of S, every rational number less than x is in S. For instance, the set of all rational nimibers less than a rational number a is a lower segment. Of two lower segments one is always a subset of the other. We may denote that aS is a subset of S' by the symbol S©S\ According to the order relation, @, every bounded set of lower segments [/S] has a least upper bound, namely the lower segment, consisting of every number in any S of [S\. If S and T are lower segments whose least upper bounds are s and t, we may define and S(g)r as those lower segments whose least upper bounds are s + t and sXt respectively. It is now easy to see that the set of lower segments contains a subset that satisfies the same conditiohs as the rational numbers, and that the set as a whole satisfies axiom K. The legitimacy of axiom K from the logical point of view is thus established, since our example shows that it cannot contradict any previous theorem of arithmetic. Further axioms might now be added, if desired, to postulate the existence of imaginary numbers, e.g. of a number x for t Cf. PiERPONT, loc. cit., pages 12-19. THE SYSTEM OF REAL NUMBERS. 13 each triad of real numbers a, b, c, such that oT^+bx+c^O. These axioms are to be justified by an example to show that they are not in contradiction with previous assumptions. The theory of the complex variable is, however, beyond the scope of this book. § 5- Axioms for the Real Number System. A somewhat more summary way of deahng with the prob- lem is to set down at the outset a set of postulates for the system of real numbers as a whole without distinguishing directly between the rational and the irrational number. Sev- eral sets of postulates of this kind have been published by E. V. Huntington m the 3d, 4th, and 5th volumes of the Transac- tions of the American Mathematical Society. The following set is due to HuNTiNGTON.f The system of real numbers is a set of elements related to ope another by the rules of addition (-I-), multiplication (x), and magnitude or order (<) specified below. Al. Every two elements a and 6 determine uniquely aa element a+b called their sum. A 2. {a+b)+c=a + ib + c). A3. (a+6) = (6+a). A 4. li a+x=a+y, then x = y. A 5. There is an element z, such that z+z=z. (This ele- ment z proves to be unique, and is called 0.) A 6. For every element a there is an element a', such that a+a'=0. M 1. Every two elements a and b determine uniquely an element ab called their product; and if a?^0 and 6?^0, then M2. iab)c=a{bc). M 3. ab = ba. M4. If ax = ay, and a^^O, then x=y. t Bulletin of the American Mathematical Society, Vol. XII, page 228. , J The latter part of M 1 may be omitted from the list of axioms, since it can be proved as a theorem from A 4 and AMI. 14 INFINITESIMAL ANALYSIS. M 5. There is an element u, different from 0, such that wu = w. This element proves to be uniquely determined, and is called 1. M 6. For every element a, not 0, there is an element a", such that aa" = l. AMI. a{h+c)=ab + ac. 1. If a^b, then either a<b or b<a. 2. If a<6, then a5^6. 3. If a<b and b<c, then a<c. O 4. (Continuity.) If [x] is any set of elements such that for B certain element b and every x,x<b, then there exists. an ele- ment B such that — (1) For every x of [x], x<B; (2) li y<B, then there is an zi of a; such that y<xi. A 1. If x<y, then a+x<a + y. M 1. If a>0 and b>0, then ab>0. These postulates may be regarded as summarizing the prop- erties of the real number system. Every theorem of real analysis is a logical consequence of them. For convenience of reference later on we summarize also the rules of operation with the symbol |a;|, which indicates the "numerical " or "abso- lute" value of x. That is, if a; is positive, la;|=a;, and if x is negative, |x|= —x. \x\ + \ymx + y\ (1) .-. I\xk\^\lxk\, (2) k-l k=l n where Ixk=xi+X2+ . . .+x„. |N-M|^k-2/| = l2/-a;|^W + |j/| (3) i2;-2/| = N-l2/l (4) (5) \X\ X \y\ ~ y If lx-t/|<ei, |2/-zl<e2, then |x-2|<ci+e2 (6) THE SYSTEM OF REAL NUMBERS. 15 If [x] is any bounded set, 5[x]-5[x]=F[|xi-X2|] (7) § 6. The Number e. In the theory of the exponential and logarithmic functions (see page 97) the irrational number e plays an important r61e. This number may be defined as follows : e=B[Er,], (1) where "^ T\'^2\'^ ' ' '"^n!' where [n] is the set of all positive integers, and n! = l-2-3...n. It is obvious that (1) defines a finite number and not infin- ity, since J?„ = l+jj+2j+ . . •+^<l+l + 2 + 2? + - • • + 2^ = ^~2^- The nimiber e may very easily be computed to any number of decimal places, as follows: ^^0 = = 1 1 1!" = 1 1 2!" = .5 1 3!" = .166666 + 16 INFINITESIMAL ANALYSIS. 1 4!~ .041666 + 1 5!~ .008333 + 1 6!~ .001388+ 1 7!~ .000198+ 1 8!~ .000024+ 1 9!" .000002 + E,= 2.7182 1 Lemma. — ^If k>e, then Ek>e--,y Proof .—From the definitions of e and f?„ it follows that '"■^*='^[(ATi)i+(Fr2)i+ •• •+(&+])!]' where [Z] is the set of all positive integers. Hence '^ik+2)...ik+t)]' or e-E,<-g^^^'e. If A>e, this gives Ek>e-^^. THE SYSTEM OF REAL NUMBERS. 17 Theorem 7. e=fil (iH — ) J, where [n] is the set of all positive integers. Proof. — By the binomial theorem for positive integers Hence E„-(l+-) = ^ (^ '-j^, ) « n*-n(n-l) . .. (n-A + 1) ,, k-2 kin'' »-, 7i*-(n-^ + l)fc •^ Tim • ^iZi kin" Hence by factoring _ / 1\" ^(A;-l)(n*-^+n*-'(n-A;+l) 1^ "^ ^(_A; - 1) (n*-i +n*-g(n-A;+l) + ... + (n-A; + l)*-^) "(A;-l);kn*-» £-2 kin" 1 ^ (A;-1)A; <n ifz k\ ' ^-('-y<-n('^a)<K• ••••<') From (a) ^„> (1+-)" d) and from (6) r+^ '^^""n' ^^^ 18 INFINITESIMAL ANALYSIS. whence by the lemma (l + l)">e-V- (3) \ nl n\ n From (1) it follows that e is an upper bound of 1\"- [(^-ri. and from (3) it follows that no smaller number can be an upper bound. Hence (('-y]-' § 7. Algebraic and Transcendental Numbers. The distinction between rational and irrational nimibers, which is a feature of the discussion above, is related to that between algebraic and transcendental numbers. A number is algebraic if it may be the root of an algebraic equation, aox"+0ix"-i + . . .+o„_ia;+a„=0, where n and ao, ai, . . . , a„ are integers and w> 0. A number is transcendental if not algebraic. Thus every rational number — n is algebraic because it is the root of the equation nx—m=0, while every transcendental number is irrational. Examples of transcendental numbers are, e, the base of the system of natural logarithms, and ;:, the ratio of the circumference of a circle to its diameter. The proof that these numbers are transcendental follows on page 19, though it makes use of infinite series which will THE SYSTEM OF REAL NUMBERS. 19 not be defined before page 71, and the function e', which is defined on page 57. The existence of transcendental numbers was first proved by J. LiouviLLE, Comptes Rendus, 1844. There are in fact an infinitude of transcendental numbers between any two num- bers. Cf. H. Weber, Algebra, Vol. 2, p. 822. No particular number was proved transcendental till, in 1873, C. Hermite (Crelle's Journal, Vol. 76, p. 303) proved e to be transcendental. In 1882 E. LiNDEMANN (Mathematische Annalen, Vol. 20, p. 213) showed that k is also transcendental. The latter result has perhaps its most interesting application in geometry, since it shows the impossibihty of solving the classical problem of constructing a square equal in area to a given circle by means of the ruler and compass. This is because any construction by ruler and compass corresponds, according to analytic geometry, to the solution of a special type of alge- braic equation. On this subject, see F. Klein, Famous Prob- lems of Elementary Geometry (Ginn & Co., Boston), and Weber and Wellstein, Encyclop'ddie der Ekmmtarrmiherruitik, Vol. 1, pp. 418-432 (B. G. Teubner, Leipzig). § 8. The Transcendence of e. Theorem 8. // c, Ci, Cz, C3, . . . , c„ are integers (or zero bvi Cj^O), then c+c.c+c,e2+ ..,+c„c»?^0 (1) Proof. — ^The scheme of proof is to find a number such that when it is multipUed into (1) the product becomes equal to a whole number distinct from zero plus a number between +1 and —1, a sum which surely cannot be zero. To find this number N, we study the series t for e*, where A; is an integer <n: t Cf. pages 71 and 99. 20 INFINITESIMAL ANALYSIS. Multiplying this series successively by the arbitrary factors i\-bi, we obtain the following equations: / k k^ \ c*l!-6i =61 -ll+ftiAU + 2+273 + -. -j; e*-2!-&2=&2-2!(l+|)+62-A;2(l+|+^+...); (k Jfc2\ / k k^ \ e*-s!-6.=6,-s!(l + j-,+2| + . . ^+-^^1)]} ft2 ^+.-Ti+ (.+i)(.+2) +---) (s+l)(s+2) For the sake of convenience in notation the numbers 61 ... 6, may be regarded as the coefficients of an arbitrary polynomial <f>{x)=bo+bix+b23^+. . .+6a, the successive derivatives of which are <l>'(x)=bi+2-b2X+. . .+s-b,-x'-\ ,i^\, XT . T (m+1)! , s! ^<">(a;)=b„.-m!+6m+i- ,, ■x+...+b.-7- r-.-a*-" The diagonal in (2) from 6-1! to b,-s \. _^.^ is obviously ^'(A), the next lower diagonal is ^"(A), etc. Therefore by adding equations (2) in this notation we obtain THE SYSTEM OF REAL NUMBERS. 21 eHV.hi+2\b2 + . . .+s\b,) = <}>'ik) + <j>"ik) + . . . + <l><'Kk) + Jb„,-k'--Rkm, (3) m-l k ^ m + l'^(m + l)(m+2) in which ftjfcm = l + zrTT + 7r-rTT7r— rirr+. • . Remembering that (f>(x) is perfectly arbitrary, we note tnat if it were so chosen that <f>'(.k)=0, <f>"ik)=0,..., .^(''-i)(A:)=0, for every A; (k = l, 2, 3, . . . , n) then equations (2) and (3) could be written in the form m-l +bp-p\ +6p+i-(p+l)!(l+^) +6..«!(l+i|+2-, + ... + (^3^). (4) A choice of (j>{x) satisfying the required conditions is ^{x) = (ao+aiX+as^ + ...+an3^y-j^:Ziy^= (p^iyT"' ^^^ where f{x) = {x-l){x-2)(x-S) . . . (x-n). 22 INFINITESIMAL ANALYSIS Every jfc (*=!, 2, . . . , n) is a p-tuple root of (5). Here p is still perfectly arbitrary, but the degree s of ^(x) is np + p-1. If 4)(x) is expanded and the result compared with <j){x)=bo+biX + . . .+b,x', it is plain that bo=0, bi=0, ..., &p_2=0, on account of the factor x^~^, and "-^ (p-1)!' " (p-1)!' ■*■' (p-1)!' where 7p, Ip+i, ...,/, are all integers. The coefficient of e*= in the left-hand member of (4) is therefore Whenever the arbitrary number p is prime and greater than Co, Np is the sum of Oo", which cannot contain p as a factor, plus other integers each of which does contain the factor p. Np is therefore not zero and not divisible by p. Further, since (p + Q! (p + l)(p+2)...(p+0 (p-l)!-r! P r! is an integer divisible by p when r^t, it follows that all the coefficients of the last block of terms in (4) contain p as a factor. Since k is also an integer, (4) evidently reduces to Np-e''=^pWkp + Ib^-k'--Rkm, THE SYSTEM OF REAL NUMBERS. 23 where W^p is an integer or zero, and this may be abbreviated to the form Np-e>'=pWkp+rkp. (6) Before completing our proof we need to show that by choosing the arbitrary prime number p sufficiently large, rtp can be made as small as we please. If a is a number greater than n, \R km] 1 + -+; F < m+1 (m + l)(?ra + 2) 2 1 + -T + W m+1 (m + l)(m + 2) + . + . <e« for all integral values of m and oik<n. \rkp\ Ibm-k^-Rkm m=l 1 I \h„\-k'^-\Rk,m\. 1=1 Since the number bm is the coefficient of x" in <f>{x) and since each coefficient of <j>ix) is numerically less than or equal to the correspanding coefficient of XP -1 (p-1)! it follows that (|ao| + |ai|a; + |a2|x2 + . . . + |a„|z»)i', !'-*p|<e''-(^ri)!(N + l«il«+---+W«")'' <(P-1)!'' 24 INFINITESIMAL ANALYSIS. where Q=a(|oo| + |ci|a + . . . + |a„|a:«) Qp / is a constant not dependent on p. The expression , _^, ^ is the pth term of the series for Qe^, and therefore by choosing p sufficiently large Vkp, may be made as small as we please. If now p is chosen as a prime number, greater than a and ao and so great that for every k, where d is the greatest of the numbers C, C\, C2, Cz, . . . , Cn, the equations (6) evidently give iV p(c + Cie + C2e= + . . . + c„e") = NpC + p{CiWij, + C2W2p + . . .+cJV„j>) + Cirip+C2r2p + . . . + c„r„p, =NpC+pW + R, (8) where W is an integer or zero and R is numerically less than unity. Since NpC is not divisible by p and is not zero, while pW is divisible by p, this sum is numerically greater than or €qual to zero. Hence Np(c+Cie+C2^ + . . .+c„e»)?^0. Hence and e is a transcendental number. THE SYSTEM OF REAL NUMBERS. 25 § 9. The Transcendence of n. The definition of the number r. is derived from Euler's formula e^"^" ^ = cos x+\/ — 1 sin x; by replacing x by n, e rN/^=-l (1) If TT is assumed to be an algebraic number, rV - 1 is also an algebraic number and is the root of an irreducible algebraic equation F{x)=0 whose coefficients are integers. If the roots of this equation are denoted by 21, 22, 23, ■ • • , Zn, then, since ^v — 1 is one of the z's, it follows as a consequence of (1) that (e^' + l)(e'= + l)(e^»+l) ...(e^" + l)=0. ... (2) By expanding (2) l + Ie'i + Ie'i+'i + Ie''+'i+h + . . .=0. Among the exponents zero may occur a number of times e.g., (c - 1) t-mes. If then Zi, Zi + Zj, Z, + Zj + Zk, ..., be designated by Xi, X2, X3, ■ ■ . , x„, the equation becomes c + e^'+e^=+.. . + e'"=0, (3) where c is a positive number at least unity and the numbers Xi are algebraic. These numbers, by an argument for which the reader is referred to Weber and Wellstein's Encyclopddie der Elemmiarmathematik, p. 427 et seq., may be shown to be the roots of an algebraic equation /(x)=ao+aiX+a2x2 + . . .+a„x" = 0, . . . (3') 26 INFINITESIMAL ANALYSIS. the coefBcients being integers and Oo 5^0 and a„ j-^O. The rest of the argument consists in showing that equation (3) is impossi- ble when xi, X2, . . . , Xn are roots of (3'). The process is analogous to that in § 8. e^>c.V.b,=bi-V.+hxu{l+^+~ + . . .) , e^..3!63 = 63-3!(l+ff+f)+63X.3(l+^+^; + ...), (4) e^..s!.6.=6..«!(l+ff+... + ^) +,^,.(l+_£L+^^ + ...) The numbers 6i, . . . , 6„ may be regarded as the coefficients of an arbitrary polynomial ^(x) = 6o+M+fe2a;2 + . . .+b^, for which <l>^'"^(x)^bm-m\+b„+i-^—:rY^-x + ... + b.j r.'X^'". J. »-i The diagonal in equations (4) from 6i-l! to b,-s\ , , is (s-1) obviously (f>'{xk), and the next lower diagonal 4>"{Xk), etc. Therefore, by adding equations (4), e**(l!6i+2!62 + . . .+s!6.) = .^'(xjt) + <^"(a;*) + . . . ■^cj>^'Kxk) + h^-Xk^Rk„„ . (5) i»»=i THE SYSTEM OF REAL NUMBERS. 27 in which 7? - 1 I ■ ^^ I ^fc^ I "*" "^m + l^(m + l)(m+2) + -'- Remembering that (f>{x) is perfectly arbitrary, let it be so chosen that <t>'{xk)=0, <A"(x,) = 0, <A'"(a;A)=0, ..., </.("- i)(xt)=0 for every xj. Equation (5) may then be written as follows : cMl!&i+2!62 + ...+s!6.) = i6„-(zt)'".i24.„ m=l + bp-pl +i>,«(p+l)!(l + j-') A choice of ^(x) satisfying the required conditions is Q^ np—i . ^p—1 4>ix) = "(2? -1)1 {ao+aix+CiX^ + . . . +a„a;")p O np-l.-cp-l - (p-1)! (^("))''- of which every xj is a p-tuple root. If <^(x) is expanded and the result compared with ^(x)=feo+6iX+. . .+b,x', it is plain that feo = 0, &i=0, . . . , 6p_2=0, on account of the factor xP~i ; and Op-i- (p_l)! ' ''p- (p_i)! ^' (p--l)! ' 28 INFINITESIMAL ANALYSIS. where /p, . . . , /, are all integers. The coefficient of e'k in (6) may now be written If the arbitrary number p is chosen as a prime number greater than a^ and a„, Np becomes the sum of aoPa„"P~i, which carnot contain p as a factor, and a number of other integers each of which is divisible by p. Np therefore is not zero and not divisible by p. ip + t)\ Further, since , _^, , — j- is an integer divisible by p when 7^', it follows that all of the coefficients of the last block of terms in (6) contain p as a factor. If then (6) is added by columns, Npe^=pan^p-^[Po+PiXk+P2Xk'' + . . .+Ps-pXk'-p] 8 + Ib„-Xk"'-Rk„,. . (7) m= 1 where Po, Pi, . . . , P^^p are integers. It remains to show tha.tIb,n-Xk"'-Rk„, can be made small at will by a suitable choice of the arbitrary p. As in the proof of the transcendence of e, it follows that ' Op |rifcp| = Ib^-Xk""- Rkm < 7 TT-j • e", m=l \P — 1^! where Q = \arJ'\a(}ao\ + \ai\a + . . . + |a„|a), and a is the largest of the absolute values of Xkik = l, . . . , n). If now p is chosen as a prime number, greater than unity, greater than ao . . . an and greater than c, and so great also that |rip| <-, it follows directly from equation (7) that THE SYSTEM OF REAL NUMBERS. 29 = Njfi+pan^p--^{PoSo+PiSr + . . .+P._pS._p) + iVfcp, (8) where \rkp\ Ihrn-XlT-Jtlcm <n' So=n, and Si = xi^+X2^+X3^+. . .+Xn\ and therefore _ ttn-l „ fln^-1 2a„_2 , ^1= — ;r~' '^2=-—^ — -— , ...,t and therefore it follows that a„"P-i/Si, a„"P~i(S2, . . . , are all whole numbers or zero. The term po„»p-i'iPiSi t = is therefore an integer divisible by p, while, on the contrary, Np and c are not divisible by p. The sum of these terms is n therefore a whole number ^+1 or ^ — 1, and since Irkp<l, the entire right-hand member of (8) is not zero, and hence (3) is not zero. Therefore — Theorem g. The number tz is transcendental. t Cf. BuRNSiDE and Panton Theory of Equations, Chapter VIII, Vol. I. CHAPTER II. SETS OF POINTS AND OF SEGMENTS. § I. Correspondence of Numbers and Points. The system of real numbers may be set into one-to-one cor- respondence with the points of a straight line. That is, a scheme may be devised by which every niomber corresponds to one and only one point of the line and vice versa. The point is chosen arbitrarily, and the points 1, 2, 3, 4, ... are at regular intervals to the right of in the order 1, 2, 3, 4, . . . from left to right, while the points -1, -2, -3, . . . foUow at regular intervals in the order 0, -1, -2, -3, . . . from right to left. The poiBts which correspond to fractional numbers are at intermediate positions as follows : t To fix our ideas we obtain a point corresponding to a par- ticular decimal of a finite number of digits, say 1.32. ? ■'. ■? ■?! ■■? ■? ? Fia. 1. Divide t he se gment 1 2 into ten equal parts. Then divide the segment 3 4 of this division into ten equal parts. The point marked 2 by the last division is the point corresponding to 1.32. If the decimal is not terminating, we simply obtain an infinite sequence of points, such that any one is to the right of all that precede it, in case of a positive number, or to the t It is convenient to think of numbers in this case as simply a notation for points. In view of the correspondence of points and numbers the num- bers furnish a complete notation for all points. 30 SETS OF POINTS AND OF SEGMENTS. 31 left in case of a negative number. The first few points of the sequence for the number a- are the points corresponding to the numbers 3, 3.1, 3.14, 3.141. This, set of numbers is bounded, 4, for instance, being an upper bound. Hence the points cor- responding to these numbers all lie to the left of the point corresponding to the number 4. To show that there exists a definite point corresponding to the least upper bound B of the set of numbers 3, 3.1, 3.14, 3.141, etc., use is made of the foUow- ing: Postulate of Geometric Continuity.— 7/ o set [x] of points of a line has a right bound, that is, if there exists a point B on the line svch that no point of the set \x] is to the right of B, then there exists a leftmost right bound B of the set [x]. If the set has a left bound, it has a rightmost left bound. The leftmost right bound of the set of points corresponding to the numbers 3., 3.1, 3.14, etc., is the point which corresponds to the number n. In the same manner it follows from the pos- tulats that there is a definite point on the line corresponding to any decimal with an infinitude of digits.! Conversely, given any point on the line, e.g., a point P, to the right of 0, there corresponds to it one and only one num- ber. This is evident since, in dividing the line according to a decimal scale, either the point in question is one of the division- points, in which case the number corresponding to the point is a terminating decimal, or in case it is not a division-point we will have an infinite set of division points to the left of it, the point in question being the leftmost right bound of the set. If now we pick out the rightmost point of this left set in every division and note the corresponding nimiber, we have a set of niunbers whose least upper bound corresponds to the point P. •f- It is not implied here, of course, that it is possible to write a decimal with an infinitude of digits, or to mark the corresponding points. What is meant is that if an infinite sequence of digits is determined, a definite number and a definite point are thereby determined. Thus V2 determines an infinite sequence of digits, that is, it furnishes the law whereby the sequence can be extended at will. 32 INFINITESIMAL ANALYSIS. The ordinary analytic geometry furnishes a scheme for set- ting all pairs of real numbers into correspondence with aJI points of a plane, and all triples of real numbers into corre- spondence with all points in space. Indeed, it is upon this correspondence that the analytic geometry is based. It should be noticed that the correspondence between num- bers and points on the hne preserves order, that is, if we have three nimibers, a, b, c, so that a<b<c, then the corresponding points A, B, C are under the ordinary conventions so arranged that B is to the right of A, and C to the right of B. It will be observed that we have not put this matter of the one-to-one correspondence between points and numbers into the form of a theorem. Rather than aiming at a rigorous demonstration from a body of sharply stated axioms, we have attempted to place the subject-matter before the reader in such a manner that he will imderstand on the one hand the necessity, and on the other the grounds, for the hypothesis. § 2 Segments and Intervals. Theorem of Borel. Definition.— A segment a b is the set of all numbers greater than a and less than b. It does not include its end-points a and b. An interval ab is the segment a b together with a and b. For a segment plus its end point a we use the notation I— — I a b, and when a is absent and b present a b. All - these nota- tions imply that a<b.-\ Sometimes we denote a segment or interval by a single letter. This is done in case it is not im- portant to designate a definite segment or interval. The set of all numbers greater than a is the infinite segment a 00, and the set of all numbers less than a is the infinite segment - 00 a. The infinite segments a oo and — oo a, together with the point a, are respectively the infinite intervals a <x and — oo a. t The notation ab,ao,ab, etc., to denote the presence or absence of end- points is du3 to G. Peano, Analisi Infinitisimali. Torino, 1893. SETS OF POINTS AND OF SEGMENTS. 33 Unless otherwise specified the expressions segment and interval will be understood to refer to segments and intervals whose end- points are finite. By means of the one-to-one correspondence of numbers and points on a line we define the length of a segment as follows: The length of a segment a b with respect to the unit segment 1 is the munber \a-b\. This definition applies equally to all segments whether they are commensurable or incommensurable with the unit segment. Definition. — A set of segments or intervals [a] covers a segment or interval t if every point of < is a point of some a. On the interval - 1 1 consider the set of points ^ . The -1 % 'A K 1 I I 1 I 1 I Fig. 2. I— I I— I I— I I 1 set of intervals -1 0, ^ 1, 4 2' • ■ • > 2^ 2^^> • ■ ■ covers I— I I— I. . the interval —1 1, because every point of —1 1 is a point of one of the intervals. On the other hand a set of segments -10,-^ 1, . . . , ^ o^^j ^*c-j does not cover the interval because it does not include the points — 1, 1, n, . . . , ^^ , . . . , or 0. In order to obtain a set of segments which does cover the inter- val, it is necessary to adjoin a set of segments, no matter how small, such that one includes -1, one includes 0, one includes 1, 2) 4l • • • The segment including 0, no matter how small it is, must include an infinitude of the points ^, and there are only a finite number of them which do not lie on that segment. It therefore follows that in this enlarged set there is a subset of segments, 34 INFINITESIMAL ANALYSIS. I— I finite in number, which includes all the points of -1 1. This turns out to be a general theorem, namely, that if any set of seg- ments covers an interval, there is a finite subset of it which also covers the interval. The example we have just given shows that such a theorem is not true of the covering of an interval by a set of intervals; furthermore, it is not true of the covering of a seg- ment either by a set of segments or by a set of intervals. I— I Theorem lo.t If an interval a b is covered by any set [a] of segments, it is covered by a finite number of segments ai, . . . , (j„ of [o]. Proof. — It is evident that at least a part of a 6 is covered by a finite number of <j's; for example, if ao is the a or one of — I the a's which include a and if b' is any point of a 6 which lies I— I in (To, then o 6' is covered by oo- Let [b'] be the set of —I I— I . all points of a b, such that a b' is covered by a finite number of a's. By Theorem 4 [b'] has a least upper bound B. To com- plete our pr oof we show (a) that B is in [b'], and (6) that B=h. (a) Let a" b" be a segment of [a] including B. Since B is the least upper bound of [b'], there is a point of [6'], b', between <i" and B. But if <ti, <t2, • • ■ . <t« be the finite set of segments I — I covering the interval a b', this set together with a" b" will I — I cover a B, which proves that S is a point of [b']. (b) If B^b, then B<b and the set ai, 02, . . . , a„ together ' . I— I with a" b", would cover an interval o c, where c is a point between B and b" ; c would therefore be a point of \b'], which is contrary to the hypothesis that B is an upper bound of [6']. Hence B=b and the theorem is proved. t This theorem is due to E. Borel, Annales de I'Ecole Nonnale Su- p^rieure, 3d series, Vol. 12 (1895), p. 51. It is frequently referred to as the Heine-Borel theorem, because it is essentially involved in the proof of the theorem of uniform continuity given by E. Heine, Die Elemente der FuTuiionetdehre, Crelle's Journal, Vol. 74 (1872), page 188. SETS OF POINTS AND OF SEGMENTS. 35 An immediate consequence of this theorem is the following, which may be called the theorem of unifarmity. I — I Theorem ii. // an interval a b is covered by a set of seg- I — I ments [a], then a b may be divided into N equal intervals such that each interval is entirely within a a. I — I Proof. — By Theorem 10 a 6 is covered by a finite set of a's, <ji, 02, ... , On. The end points of these <t's, together with a and b, are a finite set of points. Let d be the smallest distance between any two distinct points of this set. Because of the overlapping of the a's, any two points not in the same segment are separated by at least two end points. Therefore any two points whose distance apart is less than d must he on the same segment of cri, 02, ... , t7„. Now let N be such that — =rp<d, then each interval of length — r^ is contained in a a. Fig. 3. By this argument we have also proved the following: I — I Theorem 12. If an interval a b is covered by a set of seg- ments, then there is a number d such that for any two numbers Xi and X2 such that alxi<X2=& o/nd \x\ -I2I <d, there is a seg- ment a of [<t] which contains both Xi and 12. In other words, any interval of length d lies entirely vrithin some a. The sense in which these are theorems of uniformity is the I— I following. Any point x of a b, bemg witmn a segment a, can be regarded as the middle point of an interval ix of length Zi which is entirely within some a. The length Z, is in general <iifferent for different points, x. Our theorem states that a value I can be found which is effective as an l^ for every x, i.e., 36 INFINITESIMAL ANALYSIS. I— I uniformly over the interval a b. The distinction here drawn is one of the most important in rigorous analysis. It was first observed in connection with the theorem of uniform continuity ; see page 89. I— I The presence of both end points of o 6 is essential, as is shown by the following example. 1 is covered by the seg- . ~^ ^. ^ i i ments ^ 2, ^ 1, g 2' • ■ • , 2" 2^^' " ' ' ' ^^ ^^ ^ points nearer to 0, Ix becomes smaller with the lower bound 0, — I and no I can be found which is effective for all points of 1. When the end points are absent it is possible, however, to modify the notion of covering, so that our theorem remains true. This is sufficiently indicated by the following theorem, which is an immediate consequence of Theorem 10. Theorem 13. If on a segment a b there exists any set [a] of segments such that (1) [a] includes a segment of which a is an end point and a segment oj which b is an end point. (2) Every point of the segment a b lies on one or more of the segments of the set [o]. Then among the segments of the set [a] there exists a finite set of segments <ti, <t2, . . . , ct„ which satisfies conditions (1) ajid (8). The theorems which we have just proved can be generalized to space of any number of dimensions. A planar generalization of a segment is a parallelogram with sides parallel to the co- ordinate axes, the boundary being excluded. The planar gen- eralization of an interval is the same with the boundary included. The theorem of Borel becomes : Theorem 14. // every point of the interior or boundary of a parallelogram P is interior to at least one parallelogram p of a set of parallelograms [p], then every point of P is interior to at least one parallelogram of a finite subset pi . . .p„ of [p]. Proof.— Let z = 0, x=a>0, y=0, y=b>0 determine the boundary of P. Let O^yi^b. Upon the interval i of the line SETS OF POINTS AND OF SEGMENTS. 37 y=yi, cut off by P, those parallelograms of [p] that include points of i as interior points determine a set of segments [;r] such that every point of t is an interior point of one of these seg- ments ;:. There is by Theorem 10 a finite subset of [n], 7:1. . . r.n, including every point of i, and therefore a finite subset pi . . . pn of [p], including as interior points every point of i. Moreover, since the number of pi . . . p„ is finite, they include in their interior all the points of a definite strip, e.g., the points be- tween the lines y=y\—e and y=yi-\-e. y=b 1/1 V=o Fig. 4. Thus for every y\ {O^yi^b) we obtain a strip of the parallel- ogram P such that every point of its interior is interior to one of a finite number of the parallelograms [p]. These strips in- tersect the j/-axis in a set of segments that include every point of the interval b. There is therefore, by Theorem 10, a finite set of strips which mcludes every point in P. Smce each strip is included by a finite number of parallelograms p, the whole parallelogram P is included by a finite subset of [p]. The generalization of Theorems 11 and 12 is left to the reader. § 3. Limit Points. Theorem of Weierstrass. Definition.— A neighborhood or vicinity of a point a in a line (or simply a line neighborhood of a) is a segment of this fine such that a lies within the segment. We denote a line neighborhood 38 INFINITESIMAL ANALYSIS. of a point a by V{a). The symbol V*(a) denotes the set of all points of V{a) except a itself. The symbols F(oo) and 7*(oo) are both used to denote infinite segments a + oo , and F( — co ) and 'F'*( — oo) to denote infinite segments — ooa.f A neighborhood of a point in a plane (or a plane neighbor- hood of a point) is the interior of a parallelogram within which the point lies. A neighborhood of a point (a, b) is denoted by V(a, b) if (a, 6) is included and by V*{a, b) if (a, b) is excluded. Instead of the three linear vicinities V(a), V(<x>), and F( — oo) we have the following nine in the case of the plaile : V(-aj,aj) V(-oo,6) V (o, oo ,) ■ V(o, 6) V (oofoo) V (oo.fc) V(— oo, 05) V(a,- oo) V(oo,— od) Fig. 5, t This notation is taken from Pibhpont's Theory of Functions of Real Variables. It is used here, however, with a meaning slightly different from that of PlERPONT. SETS OF POINTS AND OF SEGMENTS. 39 It follows at once from a consideration of the scheme for setting the points on the Une into correspondence with all numbers that in every neighborhood of a point there is a point whose corresponding number is rational. Definition. — A point a is said to be a limit point of a set if there are points of the set, other than a, in every neighbor- hood of a. In case of a line neighborhood this says that there are points of the set in every V*{a). In the planar case this is equivalent to saying that (a, b) is a limit point of the set [x, y], either if for every F*(a) and V{b) there is an (x, y) of which x is in y*{a) and y in F(6), or if for every V(a) and 7* (6) there is an {x, y) of which x is in 7(a) and y in V*(b). Thus is a limit point of the set I ^ I, where k takes all positive integral values. In this case the limit point is not a point of the set. On the other hand, in the set 1, 1-J, 1 — 2^, . . . , 1 — oA' • • ■ > 1 is a limit point of the set and also a point of the set. In this case 1 is the least upper bound of the set. In case of the set 1, 2, 3, the number 3 is the least upper bound without being a limit point. The fimdamental theorem about limit points is the following (due to Weierstrass) : Theorem 13. Every infinite hounded set [p] of points on a line has at least one limit point. Proof. — Since the set [p] is bounded, every one of its points lies on a certain interval a b. li the set [p] has no limit point, I— I then about every point of the interval a b there is a segment a which contains not more than one point of the set [p]. By Theorem 10 there is a finite set of the segments [a] such that every point of a 6 and hence of [p] belongs to at least one of them, but each a contains at most one point of the set [p], whence [p] is a finite set of points. Since this is contrary to the hypothesis, the assimiption that there is no limit point is not tenable. 40 INFINITESIMAL ANALYSIS. It is customary to say that a set which has no finite upper bound has the upper bound + oo , and that one which has no finite lower bound has the lower bound — oo . In these cases, since the set has a point in every F*( + oo ) or in every V*( — oo ) + 00 and — 00 are also called limit points. With these con- ventions the theorem may be stated as follows: Theorem i6. Every infinite set of points has a limit point, finite or infinite. The theorem also generahzes in space of any number of dimensions. In the planar case we have: Theorem 17. An infinite set of points lying entirely within a parallelogram has at least one limit point. Theorem 17 is a corollary of the stronger theorem that fol- lows: Theorem 18. // [{x, y)] is any set of number pairs and if a is a limit point of the numbers [x], there is a value of b, finite or + 00 or —00, sv£h that for every V*{a) and V{b) there is an (x, y) of which x is in V*{a) and y is in V{b). Proof. — Suppose there is no value b finite or +00 or — 00 such as is required by the theorem. Since neither +00 nor - 00 possesses the property required of b, there is a 7* (a) and a F(oo ) and a V{-<xi) such that for every pair (x, y) of [(x, y)] whose x lies in V*{fl) y fails to he in either y(oo) or F( — 00). This means that there exists a pair of numbers M and m such that for every (x, y) whose x is in V*{a) the y satisfies the con- dition m<y<M. Further, since there exists no b such as is required by the theorem, there is for every number k on the I 1 interval m M & V{k) and a Fa*(o), such that for no (x, y) is x in Vi^{a) and y in V{k). This set of segments [F(fc)] covers the I 1 interval m M, whence by Theorem 10 there is a finite subset of [V{k)], ViQc), . . . , 7„(fc) which covers m M, and hence a finite set of corresponding Vk*{ays. Let V*(a) be a vicinity of a con- tained in every one of the finite set of 74*(a)'s and in V*{a). Hence if the x of a pair (x, y) is in V*{a), its y cannot he in one SETS OF POINTS AND OF SEGMENTS 41 Of the infinite segments W^ and -^, or in one of the finite segments V^ik), ..., V„{k), i.e., no y, corresponds to this x which IS contrary to the hypothesis. This argument covers the cases when a is + oo and when a is - « . We add the definitions of a few of the technical terms that are used in pomt-set theory, f Definition.— A set of points which includes aU its Umit points is called a closed set. A set of pomts every one of which is a Umit point of the set is called dense in itself. t A set of points which is both closed and dense in itself is called perfect. A set having no finite limit point is called discrete. A segment not including its end points is an example of a set dense in itself but not closed. If the end points are added, the set is closed and therefore perfect. The set of rational num- bers is another case of a set dense in itself but not closed. Any set containing only a finite number of points is cbsed, accord- ing to our definition. If every point of an interval a 6 is a limit point of a set [x], then [x] is everywhere dense on a b. Such a set has a point between every two points of the interval. A set which is ever3rwhere dense on no interval is called nowhere dense. All rational numbers between and 1 form an everywhere dense set. § 4. Second Proof of Theorem 15. To make the reader familiar with a style of argument which is frequently used in proving theorems which in this book are made to depend upon Theorems 10 and 14, we adjoin the fol- lowing lemma and base upon it another proof of Theorem 15. t For bibliography and an exposition in English see W. H. Young and G. C. YoTJNG, The Theory of Sets of Points. Cambridge, The University Press, t In German "in sich dicht." 42 INFINITESIMAL ANALYSIS. Lemma. — ^H3rpothesis : On a straight line there is an infinite I II 1 I 1 ... , ,„ set of intervals ai bi, O262, • • • , CLnOn, ■ • • condiiionea as follows: j . I — I I — I I — I I — I (1) Interval 02^2 lies on interval aibi, 0363 on 02^2, etc. In general o„6„ lies on a„_i&„_i. (This does not exclude the case ak=ak+\.) (2) For every interval e>0, however small, there is some n, say n«, such that \bn,—an,\ <e. Conclusion: There is one and only one point b which lies 1 — I upon every interval a„ b„. Proof. — Since the set of points ci . . . a„ . . . is bounded, we have at once, by the postulate of continuity, that this set has a leftmost right bound Ba- Similarly, the set 61 . . . 6„ . . . has a rightmost left boimd Bf It follows at once that Ba = Bi„ for if not, we get either an a point to the right of Ba, or a b point to the left of Bi when n, is so chosen that \bn.-a„,\<B,,-B^. We now give another proof for Theorem 11. Divide the in- • I— I terval a b on which aU points of [p] lie into two equal intervals. Then there is an infinite number of points [pj on at least one of these intervals which we call Ci 61. Divide this interval j" In particular the set of segments assumed in the hypothesis may be obtained by dividing any given segment into a given number of equal seg- ments, then one of these segments into the same number of equal segments and BO on indefinitely. To show that the sequential division into a num- ber of equal segments gives a set of segments satisfying the conditions of the hypothesis we have merely to show that such division gives a segment less than any assigned segment a^b^. This is equivalent to the statement tltat for every number e there is an integer n, such that — <e n a direct consequence of Theorem 3. This involves the notion that no con- stant infinitesimal exists. It may appear at first sight that a proof of this statement is superfluous. The fact is, however, as was first proved by Vebonesi:, that the non-existence of constant infinitesimals is not provable without some axiom such as the continuity axiom or the so-called Archime- dean Axiom. SETS OF POINTS AND OF SEGMENTS. 43 into two equal parts and so on indefinitely, always selecting for division an interval which contains an infinite number of points of the set [p]. We thus obtain an infinite sequence of intervals I 1 1 1 I 1 . . tti bi, a2 b2, . . . , an b„ . . . which satisfies the hypothesis of the lenrnia. There is therefore a point B which belongs to every I — I i — r I — I one of the intervals aibi, az b2, . . . , (in bn . . . , and therefore there is a point of the set [p] in every neighborhood of B. It should be noticed that the intervals in this sequence may be such that all intervals after a certain one will have, say, the right extremities in common. In this case the right extremity is the point B. Such is the sequence, obtained by decimal division, representing the number 2 = 1.99999. . . . CHAPTER III. FUNCTIONS IN GENERAL. SPECIAL CLASSES OF FUNCTIONS § I. Definition of a Function. Definition. — A variable is a symbol which represents any one of a set of numbers. A ccnstant is a special case of a vari- able where the set consists of but one number. Definition. — A variable y is said to be a single-valued Junction of another variable x if to every value of x there cor- responds one and only one value of y. The letter x is called the independent vaiiable and y the dependent variable.! Definition. — A variable y is said to be a many-valued function or multiple-valued function of another variable x if to every value of x there correspond one or more values of y. The class of multiple-valued functions thus includes the class of single-valued functions.! t This definition of function is the culmination of a long development of the use of the word. The idea of function arose in connection with coordi- nate geometry, Rene Descartes using the word as early as 1637. From this time to that of Leibnitz "function" was used synonymously with the word "power, " such as x', x', etc. G. W. Leibnitz regarded "function" as "any expression standing for certain lengths connected with a curve, such as coordinates, tangents, radii of curvature, normals, etc." JoHANN Bernoulli (1718) defined "function" as "an expression made up of one variable and any constants whatever." Leonard Euler (1734) called the expression described by Bernoulli an analytic function and introduced the notation fix) . Euler also distinguished between algebraic and transcendental functions. He wrote the first treatise on "The Theory of Functions." The problem of vibrating strings led to the consideration of trigonometric series. J. B. Fourier set the problem of determining what kind of relations <;an be expressed by trigonometric series. The possibility then under con- 44 SPECIAL CLASSES OF FUNCTIONS. 45 It is sometimes convenient to think of special values taken by these two variables as arranged in two tables, one table con- taining values of the independent variable and the other contain- ing the corresponding values of the dependent variable. Independent Variable Dependent Variable X2 2/1 2/2 If 2/ is a single- valued function of x, one and only one value of y will appear in the table for each x. It is evident that functionality is a reciprocal relation; that is, if ?/ is a function of X, then z is a function of y. It does not follow, however, that if 2/ is a single-valued function of x, then x is a single-valued function of y, e.g., y=x^. It is also to be noticed that such tables cannot exhibit the functional relation completely when the independent variable takes all values of the continuum, since no table contains all such values. Definition. — That y is & function of x (and hence that x is a function of y) is expressed by the equation y=j{x) or by x=f~^{y). If y and x are connected by the equation y=f{x), f~^(y) is called the inverse function of /(x). Thus y = x^ has the inverse function x=^±\/y. In this case, while the first function y=x^ is defined for all real values of X, the inverse function x= ±\/y is defined only for positive values of y. The independent variable may or may not take all values between any two of its values. Thus n! is a function of n where n takes only integral values. >S„, the sum of the first sideration that any relation might be so expressed led Lejedne Dirichlet to state his celebrated definition, which is the one given above. See the Encyclopadie der mathematischen Wissenschaften, II A 1, pp. 3-5; also Ball's History of Mathematics, p. 378. * 46 INFINITESIMAL ANALYSIS. n terms of a series, is a function of n where n takes only integral values. Again, the amount of food consumed in a city is a function of the number of people in the city, where the inde- pendent variable takes on only integral values. Or the inde- pendent variable may take on all values between any two of its values, as in the formula for the distance fallen from rest by a body in time t, s = -n- 'It follows from the correspondence between pairs of num- bers and points in a plane that the functional relation between two variables may be represented by a set of points in a plane. The points are so taken that while one of the two numbers which correspond to a point is a value of the independent variable, the other number is the corresponding value, or one of the corresponding values, of the dependent variable. Such representations are called graphs of the function. Cases in point where the function is single-valued are: the hyperbola referred to its asymptotes as axes \y=-) ; a straight line not parallel to the y axis {y=ax+b); or a broken line such that no line parallel to the y axis contains more than one of its points. In general, the graph of a single-valued function with a single- valued inverse is a set of points [{x, y)] such that no two points have the same x or the same y. Following is a graph of a function where the independent variable does not take all values between any two of its values. Consider Sn, the sum of the first n terms as a function of w in the series „ , 1 1 1 The numbers on the x axis are the values taken by the independent variable, while the functional relation is repre- sented by the points within the small circles. Thus it is seen that the graph of this function consists of a discrete set of points. (Fig. 6.) SPECIAL CLASSES OF FUNCTIONS. 47 The definition of a function here given is very general. It will permit, for instance, a function such that for all rational values of the independent variable the value of the function is 4 5 Fig. 6. unity, and for irrational values of the independent variable the value of the function is zero. § 2. Bounded Functions. Since the definition of function is so general there are few theorems that apply to all functions. If the restriction that fix) shall be bounded is introduced, we have at once a very im- portant theorem. Definition. — ^A function, f(x), has an upper bound for a set of valves [x] of the independent variable if there exists a finite nimaber M such that f{x) <M for every value of x in the set [x]. The function has a lower bound m if f{x)>m for every value of X in {x\. A function which for a given set of values of X has no finite upper bound is said to be unbounded on that set, or to have an upper bound + oo on that set, and if it has 48 INFINITESIMAL ANALYSIS. no lower bound on the set the function is said to have the lower bound — 00 on the set. I — 1 Theorem 19. If on an interval aba Junction has an upper bound M, then it has a least upper bound B, and there is at least I — I one value of x, x\ on a b such that the least upper bound of the function on every neighborhood of Xi contained in a b is B. Proof. — (1) The set of values of the function f{x) form a bounded set of numbers. By Theorem 4 the set has a least upper bound B. I — I (2) Suppose there were no point xi on a b such that the least upper bound on every neighborhood of x\ contained in I — I — I— I a—bisB. Then for every x of a 6 there would be a segment Ox containing x such that the least upper bound of /(x) for I— I _ values of x common to (Ji and a 6 is less than B. The set [tTj] is infinite, but by Theorem 10 there exists a finite subset [<;„] of I — 1 the set [ctJ covering a b. Therefore, since the upper bound of fix) is less than B on that part of every one of these segments of [on] which lies on a b, it follows that the least upper bound I — I _ of /(x) on a 6 is less than B. Hence the hypothesis that no point Xi exists is not tenable, and there is a point Xi such that the least upper bound of the function on every one of its I — I _ neighborhoods which lies in a 6 is B. This argument applies to multiple-valued as well as to single- valued functions. As an exercise the reader may repeat the above argument to prove the following: I — I Corollary. — If on an interval 06a function has an upper bound + 00 , then there is at least one value of x, xi on a b such that in every neighborhood of xi the upper bound of the func- tion is -I- 00 , SPECIAL CLASSES OF FUNCTIONS. 49 § 3. Monotonic Functions ; Inverse Functions. Definitions.— If a single-valued function f{x) on an interval a b is such that /(x,)</(x2) whenever a;, <X2, the function is said to be monotonic increasing on that interval. If /(xi)> /(X2) whenever xi <X2, the function is said to be monotonic de- creasing, ^'^j Fig. 7. If there exist three values of x on the interval o b, xi, X2, and X3 such that /(x2)>/(xi) and /(x2)>/(x3), while xt<X2<X3 or /(x2)</(xi) and /(x2)</(x3), while Xi<X2<X3, the function is said to be oscillating on that interval. A function which is not oscillating on an interval is called non-oscillating. It should be noticed that a function is not necessarily oscillating even if it is not monotonic. That is, it may be constant on some parts of the interval. The terms monotonic and oscillating are not convenient of application to multiple-valued functions. Hence we restrict their use to single-valued functions. Definition. — ^A function /(x) is said to have a finite niun- I — I ber of oscillations on an interval a 6 if there exists a, finite so INFINITESIMAL ANALYSIS. number of points a=xo, Xi, . . . , Xn=b, such that on each inter- val Xk-i xjfe (A; = l, 2, 3, . . . , n) f(x) is non-oscillating. It is evident that if a function has only a finite number of oscilla- I — ! tions on an interval a b and if there is no subinterval of I — I I — I a 6 on which the function is constant, then the interval a b may be subdivided into a finite set of intervals on each of 2/>sin Fig. 8. which the function is monotonic. Such a function may be called partitively monotonic (Abteilungsweise monoton). The function /(x) =sin -, for a; 5^0, and /(i) =0, for a; =0, is an example of a function with an infinite number of oscillations on SPECIAL CLASSES OF FUNCTIONS. 51 every neighborhood of a point. fix)=x sin -, for Xf^O, /(O) = 0, and f{x)=x^ sin -, for Xf^O, /(0)=0 have the above property -and also are contmuous (see page 61 for meaning of the term continuous function). There exist continuous functions which have an infinite number of oscillations on every neighborhood of every point. y — x sin i Fig. 9- The first function of this type is probably the one discovered by Weierstrass,t which is continuous over an interval and does not possess a derivative at any point on this interval (see page 150). t According to F. Klein, this function was discovered by Weierstrass in 1851. See Klein, Anwendung der Differential- und Integralrechnung auf Geometrie, p. 83 et seq. The function wa^ first published in a paper en- titled Abhandlungen aus der FunctionerUehre, Du Bois Beyuond, CreU^s .Journal, Vol. 79, p. 29 (1874). 52 INFINITESIMAL ANALYSIS. Other functions of this type have been published by Peano, Moore, and others.f These latter investigators have obtained the function in question in connection with space-filling curves. Theorem 20. If y is a monotonic function of x on the interval a b, with bounds A and B, then in turn xisa svngle-valued monotonic I — I function of y on A B, whose upper and lower bounds are b and a. Proof. — It follows from the monotonic character of 1/ as a function of x that for no two values of x does y have the same Fig. 10. I 1 value. Hence for every value of y on A JB there exists one and t G. Peano, Sut une courbe, qm remplit toiUe une aire plane, Mathematische Annalen, Vol. 36, pp. 157-160 (1890). Cesaro, Sur la representation analy- tique des regions et des courbes qui les remplisent, Bulletin des Sciences Mathi- mati/juss, 2d Ser., Vol. 21, pp. 257-267. E. H. Moore, On Certain Crinkly Curves. Transactions of the American Mathematical Society, Vol. 1, pp. 73-90 (1899). See also Steikitz, Mathematische Annalen, Vol. 52, pp. 58-69 (1899). SPECIAL CLASSES OF FUNCTIONS. 53 only one value of x. That is, a; is a single-valued function of y.f Moreover, it is clear that for any three values of y, yi, 2/2, 2/3, such that 2/2 is between ?/i and j/3, the corresponding values of ^, Xi, X2, X3, are such that x-^ is between Xi and Xa, i.e., x is a monotonic function of y, which completes the proof of the theorem. CcroUary.— If a function /(x) has a finite number ^ of oscil- lations and is constant on no interval, then its inverse is at most (A + l)-valued. For example, the inverse of y = x^ is double- valued. § 4. Rational, Exponential, and Logarithmic Functions. Definitions. — The symbol a"", where wi is a positive integer and a any real number whatever, means the product of m factors a. This definition gives a meaning to the symbol y=a„x'"+am-ix"'-^ + . . .+aix + ao, •where oo . . . a^ are any real numbers and m any positive inte- ger. In this case y is called a rational integral function of x or a polynomial in x.t In case amX"'+am_iX"'~^+ . . . +ai-x+ao ^~ 6„x" + 6„_iX"-»+ . . . +6i-x+6o ' m and n being positive integers and a* {k=0, . . .m) and bi (Z = 0, . . . n) being real numbers, y is called a rational function of X. If yn+yfi-lR^(x)+y^-m2{x)+ ... +yRn.l{x) +Rnix) =0, where Ri{x) . . . R„(x) are rational functions of x, then y is said to t it is clear that the independent variable y of the inverse function may not take on all values of a continuum even if x does take on all such values. % The notion of polynomial finds its natural generalization in that of a power series y=c^ + c-x + C2-x'+ . . . +c„i"+ . . . For conditions under which a series defines y as a. function of x see Chapter IV, § 3. 64 INFINITESIMAL ANALYSIS be an algebraic function of x. Any function which is not algebraic is transcendental. The symbol a', where a; = — , m and n being positive integers and a any positive real number, is defined to be the nth root of the mth power of a. By elementary algebra it is easily shown that o*i.a=^=o*'+*» and {a''^)'"=a''^'". If y=a', then 2/ is an exponential function of x. At present this function is defined only for rational values of x. Fia. 11. Theorem 21. The function a' for x on the set \ — \ is a monotonic increasing function if l<a, and a monotonic decreasing function ifO<a<l. Proof. — (a) For integral values of x the theorem is obvious. (6) If xi = — and X2 =—, where — >— , then «i ni Til Wi SPECIAL CLASSES OF FUNCTIONS. 55 c*'<a*' if o>l and a*'>a*» if a<l. The proof of this follows at once from case (a), since a"i = \a»ij (by definition and ele- mentary algebra) and an, = ^a»ij . (c) If xi= — and X2= — , where — < — , we have Til n-z' rii 712 o"> =a"i"2 and a"2 =o"«"i, where mi 712 >in2-ni, which reduces case (c) to case (&). This theorem makes it natural to define a", where a > 1 and x is a positive irrational number, as the least upper bound of all r "1 m numbers of the form La" J, where — is the set of all posi- tive rational nimibers less than x, i.e., a'^=Bta^J. It is, however, equally natural to define a" as ^Lo^J, where I I is the set of all rational numbers greater than x. We shall prove that the two definitions are equivalent. Lemma. — If [x] is the set of all positive raiional numbers, then 5[a^] = l ifa>l and B[a='] = l ifa<l. Proof. — We prove the lemma only for the case o>l, the argument in the other case being similar. If x is any positive 7?l 1 rational number, — , then the number - is less than or equal i_ ^ fl 1 to x, and since a' is a monotonic function, a'*<a^. But - I is a subset of I -~ I- Hence where [n] is the set of all positive integers. 56 INFINITESIMAL ANALYSIS. If 5La»J were less than 1, then there would be a value, ni, of n such that a»i<l. This implies that a<l, which is con- trary to the hypothesis. On the other hand, if 5La" J > 1, there is a number of the form 1+e, where e>0, such that l + e<a" for every n. Hence {l+e)"<a for every n, but by the binomial theorem for integral exponents (l+e)''>H-ne, and the latter expression is clearly greater than a if a n>-. e Since 5La"J cannot be either greater or less than 1, Theorem 22. // x is any real number, and \~\ the set of all rational numbers less than x, and \-\the set 0} all rational numbers greater than x, then 5La"J = BLaO ifa>l, Bla'^j^Bla'^J ifO<a<l. Proof. — We give the detailed proof only in the case a>l, — 2 n] is zero, _ ai-a„J=Blai[l-a«~ vJJ is also zero. Now if B[a'^]^B[af'], SPECIAL CLASSES OF FU^CT10I^S. 57 Since as is always greater than a", Bla^j-Bla^j = £>0. But from this it would follow that p m a9 -a" is at least as great as e, whereas we have proved that _ a«-o"J=0. Hence 5La»J = sLa9j ifa>l. Definition. — In case x is a positive irrational number, and I — I is the set of all rational numbers greater than x, and [— is the set of all rational numbers less than x, then a» = Bla^J - 5L0 » J if a > 1 and a*=5La«] = B[a"] ifO<o<l. Further, if a; is any negative real number, then a'= — ; and a° = l. a' . Theorem 23. The function a* is a monotonic increasing func- tion ofxifa>l, and a monotonic decreasing function if 0<a<l. In both cases its upper bound ts + <» and Us lower bound is zero, the function taking all values between these bounds; further, ax,.^x,=ai,+i2 and {d'^Y^ = a^'-''K The proof of this theorem is left as an exercise for the reader. The proof is partly contained in the preceding theorems and 58 INFINITESIMAL ANALYSIS. involves the same kind of argument about upper and lower bounds that is used in proving them. Definition. — The logarithm of x (x>0) to the base a{a>0) is a number y such that a^=x, or a^°^'==x. That is, the func- tion logo X is the inverse of a''- The identity gives at once logo xi+ logo X2= logo (xi ■ X2) , and (a*')^'=a''"^^ gives Xi-logoX2=logaX2'^. By means of Theorem 20, the logarithm logaX, being the inverse of a monotonic function, is also a monotonic function, increasing if 1 < a and decreasing if 0< a < 1. Further, the func- tion has the upper bound + 00 and the lower bound — 00 , and takes on all real values as x varies from to +00 . Thus it follows that f or i< a, 1 < b, B(\ogb x) =log6 a=log6 (Bx). By means of this relation it is easy to show that the function x", (a;>0) is monotonic increasing for all values of a, a>0, that its lower bound is zero and its upper bound is + w , and that it takes on all values between these bounds. The proof of these statements is left to the reader. The general type of the argument required is exemplified in the following, by means of which we infer some of the properties of the function x". If xi < xz, then l0g2 Xi < log2 X2, and Xi • log2 xi < X2 • log2 X2, and log2 xi *' < log2 X2*». .". Xi*><X2*». SPECIAL CLASSES OF FUNCTIONS. 59 Hence i*, (x>0) is a monotonic increasing function of x. Since the upper bound of x ■ log2 x = log2 x^ is + <» , the upper bound of X* is +00. The lower bound of x* is not negative, since x>0, and must not be greater than the lower bound of 2=^, since if x<2, x''<2*; since the lower bound of 2* is zerot the lower bound of x* must also be zero. Further theorems about these functions are to be found on pages 64, 81, 97, 123, and 160. •f The lower bound of a' is zero by Theorem 23. CHAPTER IV. THEORY OF LIMITS. §1. Definitions. Limits of Monotonic Functions. Definition. — If a point a is a limit point of a set of values taken by a variable x, the variable is said to approach a upon the set; we denote this by the symbol x = a. a may be finite or +00 or — 00 . In particular the variable may approach a from the left or from the right, or in the case where a is finite, the variable may take values on eacl^ side of the limit point. Even when the variable takes all values in some neighborhood on each side of the limit point it may be important to consider it first as taking the values on one side and then those on the other. Definition. — A value b (6 may be + oo or - oo or a finite number) is a value approached by f{x) as x approaches a if for every V*{a) and V(b) there is at least one value of x such that x is in V*{a) and f(x) in V(b). Under these conditions f(x) is also said to approach 6 as x approaches a. Definition. — If b is the only value approached as x ap- proaches a, then b is called the limit of f(x) as x approaches a. This is also indicated by the phrase "f{x) converges to a unique limit b as X approaches a," or "f{x) approaches b as a limit," or by the notation L fix) =b. x—a The function f{x) is sometimes referred to as the limitand. The set of values taken by x is sometimes indicated by the sym- bol for a limit, as, for example, 60 THEORY OF LIMITS. 61 L j(x)=b or L f{x)=b or L f{x)=b. x>a x<a x\[x] x^a x±a z±a The first means that x approaches a from the right, the second that X approaches a from the left, and the third indicates that the approach is over some set [x] otherwise defined. Definition. — If f{x) is single-valued and converges to a finite limit as x approaches a and L/(x)=/(a), x-a then /(x) is said to be continuous at x=a. By reference to § 3, Chapter II, the reader will see that if 6 is a value approached by fix) as x approaches a, then (a, b) is a limit point of the set of points (x, /(x)). Theorem 18 therefore translates into the following important statement: Theorem 24. If /(x) is any function defined for any set [x] of which a is a {finite or +« or — 00 ) limit point, then there is at least one value (finite or +« or -00) approached by f{x) as x ap- proaches a. Corollary. — If fix) is a bounded function, the values ap- proached by fix) are all finite. In the light of this theorem we see that the existence of L fix) x=a simply means that fix) approaches only one value, while the non-existence of Lfix) means that fix) approaches at least two values as x approaches a. In case fix) is mono tonic (and hence single-valued), or more generally if fix) is a non-oscillating function, these ideas are particularly simple. We have in fact the theorem: Theorem 25. If fix) is a non-oscillating function for a set of values [x]<a, a being a limit point of [x], then as x approaches a 62 INFINITESIMAL ANALYSIS. from the left on the set [x], f{x) approaches one and only one value b, and if f{x) is an increasing function, b = Bf{x) for X on [x], whereas if f{x) is a decreasing function, h = B}{x) for X on [x]. Proof. — Consider an increasing non-oscillating function and let h = Bf{x) for X on [x]. In view of the preceding theorem we need to prove only that no Value 6V6 can be a value approached. Suppose h'>h; then since Bf{x) =b, there would be no value of fix) between b and b', that is, there would be a V(b') which could contain no value of fix) , whence b'>b 'm not a value approached. Sup- pose b'<b. Then take b'<b"<b, and since Bf{x)=b, there would be a value Xi of [x] such that fixi)>b". If xi<x<a, then b" <fixi)^fix), because fix) cannot decrease as x in- creases. This defines a V*(a) and a V{b') such that if x is in y*(a), fix) cannot be in V{b'). Hence b' <b is not a value approached. A like argument applies if fix) is a decreasing function, and of course the same theorem holds if x approaches a from the right. It does not follow that Lfix)=Lfix), x<a x^a x±a x—a nor that either of these limits is equal to /(a) . A case in point is the following: Let the temperature of a cooling body of water be the independent variable, and the amount of heat given out in cooling from a certain fixed temperature be the dependent variable. When the water reaches the freezing- THEORY OF LIMITS. 63 point a great amount of heat is given off without any change in temperature. If the zero temperature is approached from below, the function approaches a definite limit point k. and if the temperature approaches zero from above, the function Hea« Temp. FiQ. 12. approaches an entirely different point A-'. This function, how- ever, is multiple-valued at the zero point. A case where the limit fails to exist is the following: The function y=anl x (see Rg. S, page 50) approaches an infinite number of values as J approaches zero. The value of the fimction will be alter- D&tely 1 and —1, as j = -. ^, -r-, etc., and for all values of X between any two of these the function will take all values between 1 and —1. Clearly every value between 1 and -1 is a value approached as r approaches zero. In like manner 64 INFINITESIMAL ANALYSIS. 1/ =- sin - approaches all values between and including + » and - 00 , cf. Fig. 13. tslni Fig. 13. The functions a*, logo x, x° defined in § 4 of the last chapter are all monotonic and all satisfy the condition that L f{x)=fia)=L fix), x>a x<a z£a x—a at all points where the functions are defined. These functions are therefore all continuous. THEORY OF LIMITS. 66 § 2. The Existence of Limits. Theorem 26. A necessary and sufficient condition ■\ that f(i) shall converge to a unique limit h as x approaches a, i.e., that L f{x)=b, is that for every V(b) there shall exist a V*{a) such that for every xin r*{a), fix) is in r(b). Proof. — (1) The condition is necessary. It is to be proved that if L fix) =b, then for every T'(6) there exists a F*(a) such that for every x in V*(a) the corresponding f{x) is in T'(b). If this conclusion did not follow, then for some V{b) every V*(a) w-ould contain at least one x' such that /(.j-') is not in T'(6). There is thus defined a set of points [x'] of which a is a limit point. By Theorem 20 f{x) would approach at least one value h' as X approaches a on the set [/]. But by the definition of [/], b' is distinct from h. Hence the hypothesis would be con- tradicted. (2) The condition is sufficient. We need only to show that if for every. V(b) there exists a V*{a) such that for every X in V*ia) the corresponding fix) is in V(b). then f{x) can approach no other value than b. If b'j^b, then there exists a 1(6') and a Vib) which have no point in common. Now if V*ia) is such that for every x of T'*(a), f{x) is in Vib), then fThis means: (o) If L f(x) = b, then for e\-ery V(5) there exists a T"*(o), as specified by the theorem. (6) If for e^-ery r(6) there exists a T"*(a) as specified, then L /i,j) = b. A condition is necessary for a certain conclusion if it can be deduced from that conclusion ; a condition sufficient for a conclusion is one from which the conclusion can be deduced, A man sufficient for a task is a man -who can perform the task, \riiile a man necessary for the task is such that the task cannot be performed without him. 66 INFINITESIMAL ANALYSIS. for no such x is f{x) in V{b') and hence b' is not a value approached. The reader should observe that this proof applies also to multiple-valued functions, although worded to fit the single- valued case. It is worthy of note that in case 6 is a finite num- ber, our theorem becomes : A necessary and sufficient condition that Lf{x)=h xiza is that for every £>0 there exists a Vt*(a) such that for every x mV*(a), \f{x)-b\<B. In case a also is finite, the condition may be stated in a form which is frequently used as the definition of a limit, namely : L fix) =b means that for every e>0 there exists a 5, >0 sv^h that if \x—a\<d, and Xy^a, then |/(x)— 6|<£.t Theorem 27. A necessary and sufficient condition that f{x) ■shall converge to a finite limit as x approaches a is that for every e>0 there shall exist a V*{a) such that if Xi and X2 are any two values of x in V,*{a), then |/fe)-/(x2)|<e. Proof. — (1) The condition is necessary. If Lf{x)=b and h is finite, then by the preceding theorem for every ^>0 there ■exists a V *(o) such that if xi and X2 are in V *{a), then l/(^i)-6|<| and \fix2)-b\<^, from which it follows that |/(Xl)-/fe)|<£. tThe E subscript to ^. or to F.*(o) denotes that d, or V,*{a') is a func- tion of e. It is to be noted tha' inasmuch as any number less than S is effective as dt, dt is a multiple- valued function of e. THEORY OF LIMITS. 67 (2) The condition is sufficient. If the condition is satisfied, there exists a V*{a) upon which the function j{x) is bounded. For let 7 be some fixed number. By hypothesis there exists a 'V*{a) such that if x and Xo are on y*(o), then |/(x)-/(xo)|<i. Taking Zo as a fixed number, we have that /(xo)-£</(x)</(xo)+7 for every x on V*{a). Hence there is at least one finite value, 6, approached by /(x). Now for every e > there exists a y,*(a) such that if xi and X2 are any two valves of x in V*{a)^ |/(xi) — /(X2)| < £. Hence by the definition of value approached there is an x, of 7.* (a) for which W.)-'b\<^ (a) and |/(x,)-/(x)l<£ (6) for every x of F.*(a). Hence, combimng (a) and (6), for every x of 7 *(o) we have l/(x)-6|<2£, and hence by the preceding theorem we have L/(x)=6. x-a In case a as well as 6 is finite. Theorem 27 becomes: A necessary and sufficient condition thai Lfix) x±a shaU exist and be finite is that for every e>0 there exists a d.>0 such that \i{Xx)-KXi)\<^ 68 INFINITESIMAL ANALYSIS. for every Xi and X2 such that XiT^a, XzT^a, \xi—a\<d., \x2-a\<8,. In case a is + 00 the condition becomes : For every e > there exists a N.>0 such that \Kxi)-f{x2)\<e for every xi and X2 such that Xi>N„ X2>N,. The necessary and sufficient conditions just derived have the following evident corollaries : Corollary 1. The expression Lfix)=b, x~a where b is finite, is equivalent to the expression Lif{x)-b)=0, and whether h is finite or infinite L fix) =6 is equivalent to L (-fix)) = —6. Corollary 2. The expressions L/(a;)=0 and L |/(a;)|=0 are equivalent. Corollary 3. The expression Lfix)=b is equivalent to Lfiy+a)=b, »=o where y+a=x. THEORY OF LIMITS. Corollary 4. The expression L/(i)=6 x<a i = o IS equivalent to jjHh- where 2= x—a The reader should verify these corollaries by writing down the necessary and sufficient condition for the existence of each limit. The following less obvious statement is proved in detail for the case when b is finite, the case when b is + « or — « being left to the reader. Corollary 5. If Lfix)=b, then L|/(x)| = |b|. Xza Proof. — By the necessary condition of Theorem 26 for every e there exists a 7.* (a) such that for every Xi of V*{a) \f{x^)-b\<e. If /(Ji) and b are of the same sign, then ||/(xi)|-|6|| = |/(xO-6|<«, and if /(xi) and b are of opposite sign, then |l/(xi)i-|b||< 1/(^1) -M<^. Hence, by the sufficient condition of Theorem 26, L \fix)\ x=a exists and is equal to |6|. 70 INFINITESIMAL ANALYSIS. Corollary 6. If a function /(x) is continuous at x=a, then |/(x)| is continuous at a;=a. It should be noticed that L|/(x)| = |6| is not equivalent to L fix) =b. Suppose j{x) = 4-1 for all rational values of x and f{x) = — 1 for all irrational values of x. Then L |/(a;)| = +1, but L f{x) does x=a x=a not exist, since both + 1 and — 1 are values approached by f{x} as X approaches any value whatever. Definition. — Any set of numbers which may be written [i„], where n=0, 1,2, ...,K, or n=0, 1, 2, ...,«,.. ., is called a Sequence. To the corollaries of this section may be added a corollary- related to the definition of a limit. Corollary 7. If for every sequence of numbers [i„] having a as a limit point, L f{x) = h, then L f{x)=b. x|[ln] x-a x= a Proof. — In case two values b and 6i were approached by f{x) as X approaches a, then, as in the first part of the proof of Theorem 26, two sequences could be chosen upon one of which fix) approached b and upon the other of which fix) approached bi. § 3. Application to Infinite Series. The theory of limits has important apphcations to infinite series. An infinite series is defined as an expression of the form THEORY OF LIMITS. 71 00 i'ot=ai +02+03+ . . . +a„+ ... If Sn is defined as n Ol + ... +an = -^ dkt k=l n being any positive integer, then the sum of the series is defined as LSr,=S no- 00 if this limit exists. If the Umit exists and is finite, the series is said to be conver- gent. If (S is infinite or if <S„ approaches more than one value as n approaches infinity, then the series is divergent. For exam- ple, S is infinite if lafc = l + l + l + l..., and Sn has more than one value approached if ioi = l-l + l-l + l... It is customary to write Rn = S — Sn. A necessary and sufficient condition for the convergence of an infinite series is obtained from Theorem 27. (1) Fm- every e>0 there exists an integer N. such thai if n>N, and n'>N., then \S„-Sn'\<e. This condition immediately translates into the following form: 72 INFINITESIMAL ANALYSIS. (2) For every £>0 there exists an integer N. swh that if n>N„ then for every k |o„+o„+i + . . .+a„+fc|<e. Corollary. — If 2 ot is a convergent series, then L 04=0. k=l *=« Definition. — A series 2'ai=Oo+ai + . . .+On+. • • is said to be absolutely convergent if |ap| + |ai I + . . . + |a„| + . . . is convergent. Since |a„ + 0„+i+. . .+0„+fc|<|a„| + l0n+l|+. . .Iffln+ltij the above criteria give Theorem 28, A series is convergent if tt is absolutely conver- gent. 00 Theorem 29. If I bkis a convergent series all of whose terms k=0 00 are positive and I ak is a series such thai for every k, \ak\S)k, k=Q then I o* k-O is absolutely convergent. Proof. — By hypothesis k=0 k-0 THEORY OF LIMITS. 73 n Hence I \ak\ is bounded, and being an increasing function of n, the series is convergent according to Theorem 25. This theorem gives a useful method of determining the con- vergence or divergence of a series, namely, by comparison with a known series. Such a known series is the geometric series a+ar+ar^+ . . . +ar^+ . . . , where 0<r<l and a>0. In this series „ l_rn+l a which shows that the series is convergent. Moreover, it can easily be seen to have the sum :; — . •^ l-r If rf 1, the geometric series is evidently divergent. This result can be used to prove the "ratio-test " for convergence. Theorem 30. // there exists a number, r, 0<r<l, such that an ^ a„_il for every integral value of n, then the series ai+a2+ . . . +an + (1) is absolutely convergent. If a„-i >1 for every n, the series is divergent. Proof. — ^The series (1) may be written 02 02 03 . , az an ,n~. Oi+Oi-+ai + ■•• +°'^:r ■•■n — . ■ • (2) 74 and if On Ctn-l INFINITESIMAL ANALYSIS. <r, this is numerically less term by term than Oi+Oir+Oir^ . . .+air" + (3) and therefore converges absolutely. If dn ^l,a„^i for every n; hence, by the corollary, page 72, (1) is divergent. Nothing is said about the case when <1, but L an_ dn-l It is evident that the ratio test need be applied only to terms beyond some fixed term a„, since the sum of the first n terms ai+a2 + . . .+o„ may be regarded as a finite number Sn and the whole series as i.e., a finite number plus the infinite series an+i+an+2 + . • • § 4. Infinitesimals. Computation of Limits. Theorem 31. A necessary and sufficient condition thai L}{x)=b x±a is that for the function e(x) defined by the equation f{x) =b + e(x) L e{x)=0. THEORY OF LIMITS. 75 Proof.— Take e{x)=f{x)-b and apply Theorem 26. A special case of this theorem is: A necessary and sufficient am- dition for the convergence of a series to a finite value b is that for every e>0 there exists an integer N. such that if n>N., then \Rn\<e. Definition.— A function f{x) such that Lf{x)=0 is called an infinitesimal as x approaches o.f Theorem 32. The sum, difference, or product of two infinitesi- mals is an infinitesimal. Proof. — Let the two infinitesimals be /i(x) and fzix). For every £, 1> £>0, there exists a Vi*{a) for every x of which and a F2*(o) for every x of which Ux)\<^. Hence in any V*(a) common to Vi*{a) and V2*{a) \hix)+f2ixMfi{x)\ + Ux)\<e, \h(x)-f2(xmflix)\ + \f2ix)\<,, \hix)-f2{x)\ = \fi(x)\-\f2{x)\<e. From these inequalities and Theorem 26 the conclusion follows. Theorem 33. // f{x) is bounded on a certain V*{a) and £(i) is an infinitesimal as x approaches a, then £{x) •f{x) is also an infinitesimal as x approaches a. t No constant, however small if not zero, is an infinitesimal, the essence of the latter being that it varies so as to approach zero as a limit. Cf. Goursat, Cours d' Analyse, tome I, p. 21, etc. 76 INFINITESIMAL ANALYSIS. Proof. — By hypothesis there are two numbers m and M, such that M>f{x)>m for every x on V*{a). Let k be the larger of |m| and \M\. Also by hypothesis there exists for every e a V,*(a) within V*{a) such that if x is in V*ia), then or A;|£(x)|<£. But for such values of x \f{x)-E{x)\<k-\E(x)\<e, and hence for every e there is a V*ia) such that for x an y,*(o) \Kx)-c{x)\<e. Corollary. — If }{x) is an infinitesimal and c any constant, then c-f{x) is an infinitesimal. Theorem 34. 7/ L/i(x)=6i ond Lf2{x)=b2, x~a x~a bi and 62 6et7ig /inite, then L\fi{x)±f2(x)\=bi±b2, ... (a) L\h{x)-f2{x)}=bi-b2; (/?) andx/6.^0, lJ^^J^^ (,) ^ Proof. — According to Theorem 31, we write /i(a;)=6i + £i(z), h{x)=b2 + B2ix), THEORY OF LIMITS. 77 where €i(a;) and £2(2;) are infinitesimals. Hence fi{x)+f2{x)=bi+b2 + siix) + s2ix), . . . (a') fi{x)-J2{x)=bi-b2 + bi-£2{x)+b2-ei{x) + ei{x)-e2{x). . (/?') But by the preceding theorem the terms of (a') and (/3') which involve £1(2) .and e2{x) are infinitesimals, and hence the con- clusions (a) and (/?) are established. To establish (;-), observe that by Theorem 26 there exists a V*{a) for every x of which 1/2(2) -&2I <|62| and hence upon which /aCx) t-^O. Hence fijx) ^ bi + sijx) J}i b2eiix) -biezjx) hix) 62 +£2(2) 62 62162+ £2(X)} ' the second term of which is infinitesimal according to Theorems 32 and 33. Some of the cases in which 61 and 62 are ± 00 are covered CO by the following theorems. The other cases (oo -oo, — , -, etc.), are treated in Chapter VI. Theorem 35. i/ ]2{x) has a lower bound on some V*{a), and if L/,(2)=+oo, 1=0 then L i/2(a;)+/i(x)} = + oo. Proof.— Let M be the lower bound of /2(x). By hypothesis, for every number E there exists a VE*{a) such that for x on VE*ia) U{x)>E-M. Since hix)>M, this gives hix)+J2{x)>E, which means that /i(x) +J2{x) approaches the limit + 00 . 78 INFINITESIMAL ANALYSIS. Theorem 36. 7/ L/i(x) = + oo or —00, and if f2(x) is such that for a F*(a) /2(x) has a lower bound greater than zero or an upper boundless than zero, then L {/i(x) 72(2;) } is definitely infinite; x±a i.e., if /2(x) has a lower bound greater than zero and Lfi{x) = + co, z = a then L {fi{x) ■f2{x) ! = +<», etc. x~za Proof. — Suppose /2(x) has a lower bound greater than zero, say M, and that L fi{x) = + qo . Then for every E there exists a x~a - E Te*{o) within V*{a) such that for every Xi of F£*(a), fi{xi) > t^, and therefore fi{xi)-f2{xi)jfi{xi)-M>E. Hence by the defini- tion of Umit of a function Lj/i(x) ■/2(a;)! =+». If we consider the case where /2(x) has an upper bound less than zero, we have in the same manner L {/i(x) •/2(a;) t = — <» ■ Similar state- x±a ments hold for the cases in which L /i(x) = — 00 . Corollary. — If /2(x) is positive and has a finite upper bound .andL/i(x) = +oo, thea iv ■ , ^ = + 00 . X±af2{x) Theorem 37. If L /(x) = + (», then L 77-: = 0, and there is' a x~a x~aJ\X) ■vicinity V*{a) upon which /(x) >0. Conversely, if L /(x) =0 and x=a there is a V*{a) upon which /(x) > 0, then L jr— = + 00 . Proof. — If L f{x) = + 00 , then for every e there exists a x-±a V*{(i) such that if x is in V*{a), then /(^)>7 THEORY OF LIMITS. 79 and 77-T<«- since both f{x) and 7^ are positive. Again, if L /(x) =0, then for every e there is a 7.* (a) such x=.a that for X in V*{a), |/(x)|<e or — ->- (/(x) being positive). Hence L 7^-r = + 00 . Corollary 1. If /i(x) has finite upper and lower bounds on some V*{a) and L /2(x) = + « or - 00 , then Corollary 2. If /2(x) is positive and /i(x) has a positive lower bound on some V*{a) and L f2{x)=0, then x£o £( . , . = +00. x=a/2(a;) Theorem 38 (change of variable). If (1) L/i(j)=6i and L/2(2/)=&2 wfecn y takes all values of /i (x) corresponding to valves of x on some V*{a), and if (2) /i{j) f^bi for X on V*{a), then L /•,(/, (x)) =62. x-a 80 INFINITESIMAL ANALYSIS. Proof.— (a) Since L /zCj/) =62, for every 7(62) there exists a F*(6i) such that if y is in V*(bi), jiiy) is in V{hi). Since L fi{x) =61, for every F(6i) there exists a 7*(a) in F*(a) such xda that if a; is in y*(o), /i(x) is in F(6i). But by (2) if x is in V*{a), fi{x)^bi. Hence (/?) for every V*(bi) there exists a V*{a) such that for every x in V*(a), /i(x) is in V*{bi). Combining statements (a) and (/3) : for every 1^(62) there exists a F*(a) such that for every x in V*{a) fi{x) is in F*(6i), and hence fzifix)) is in F(62). This means, according to Theo- rem 26, that Lf2ih{x))^b2. x~a Theorem 39. If L /i(x) =6 and L f2(y) =hQ>)} where y takes x±a yxb all valves taken by /i(x) for x on some V*ia), then L/2(/i(x))=/,(6). x~a Proof. — The proof of the theorem is similar to that of Theorem 38. In this case the notation /2(6) implies that 6 is a finite number. Thus for every ej there exists a V,*{a) entirely within V*{a) such that if x is in V,*{a), |/i(x)-fe|<n. Furthermore, for every £2 there exists a 5,, such that for every 2/, j/7^6, \y-b\<dE2, \f2{y)-f2{b)\<e2. But since Ifziy) -f2{b)\=0 when y = b, this means that for all values of y (equal or unequal to b) such that \y—b\<d,^, 1/2(2/) -/2(b)|<e2. Now let e, =5„; then, if i is in V.*{a), it follows that 1/1 (z) -b\<d„ and therefore that |/2(/l(x))-/2(6)|<S2. Hence L f2(f,{x))=f2{b). THEORY OF LIMITS 81 Corollary 1. If /^(x) is continuous a,t x=a, and f-ziy) is con- tinuous at 2/ = /i(a), then /2(/i(x)) is continuous at x = o. Corollary 2. If ^v 0, /(x)^0, and L /(x) = b, then under the convention that oo'= = oo if k>0 and oo*:=0ifjfc<0. Corollary 3. If OO and /(x)>0 and 6>0 and Lf{x)=b, ^'^ Llog,/(x)=log,6, under the convention that log,. ( + w ) = + oo and log„ = - oo . The conclusions of the last two corollaries may also be ex- pressed by the equations L(/(x))* = (L/(x))* x—a ar^a and log, L/(x)=L log, '(x). x'-a x±a Corollary 4. If L (/(x))* or L log /(x) fails to exist, then L /(x) does not exist. § 5. Further Theorems on Limits. Theorem 40. 7/ f{x):^b for all valves of a set [x] on a certain V*{a), then every value approached by fix) as x approaches a is less than or equal to b. Similarly if fix)^b for all values of a set [x] on a certain F*(o), then every valve approached by /(x) as x approaches a is greater than or equal to b. Proof. — If f{x)%b on V*{a), then if b' is any value greater than b, and V{b') any vicinity of b' which does not include b, there is no value of x on V*(a) for which /(x) is in V{b'). Hence V is not a value approached. A similar argument holds for the case where f{x)^b. 82 INFINITESIMAL ANALYSIS. Corollary 1. If /(x)^0 in the neighborhood of x=c, then if L fix) exist, L /(x)^0. Corollary 2. If fiixj^fzix) in the neighborhood of x=a, then L /i(x)2 L U^) x±a i£o if both these Umits exist. Proof.— Apply Corollary 1 to fi(x)-f2{x). Corollary 3. If t\{x)'^J2{x) in the neighborhood of x=a, then the largest value approached by /i(x) is greater than or equal to the largest value approached by /2(x). Corollary 4. If /i(x) and ^(x) are both positive in the neighborhood of x=a, and if /i(x)^/2(x), then if L /i(x)=0, it follows that L/2(x)=0. Theorem 41. // [ucf] is a subset of [x], a being a limit point of [x'], and if L f{x) exists^ then L /(a/) exists and x-±a x±a L f{x) = L fix').^ x~a x'±a Proof. — By hypothesis there exists for every V{b) a V*{a) such that for every x of the set [x] which is in V*{a), f{x) is in F(6). Since [x'] is a subset of [x], the same V*{a) is evidently efficient for x on [x']. In the statement of necessary and sufficient conditions for the existence of a limit we have made use of a certain positive multiple-valued function of e denoted by 8,. If a given value is effective as a d„ then every positive value smaller than this is also effective. Theorem 42. For every e for which the set of valves of d, has an upper bound there is a greatest d,. t The notation /(i') is used to indicate that x takes the values of the 8et[x']. THEORY OF LIMITS. 83 Proof. — ^Let B[d.] be the least upper bound of the set of values oi d, for a particular e. If x is such that \x—a\ < B[8,], then there is a d, such that |x-a|<^,. But if \x-a\<d„ |/(z) -b\ < e. Hence, if \x-a\<B[d.l \f{x) -b\ < e. Theorem 43. The limit of the least upper bound of a function fix) on a variable segment a x,a<x, as the end point apjrroaches a, is the least upper bound of the valv£s approached by the function as X approaches a from the right. Proof.^Let / be the least upper bound of the values ap- proached by the function as x approaches a from the right, and let b{x) represent the upper bound of fix) for all values of x on a X. Since Bfix) on the segment a Xi is not greater than Bfix) on a segment a X2 if xi lies on a x^, bix) is a non-oscillat- ing function decreasing as x decreases. Hence L bix) exists by Theorem 21; and by Corollary 3, Theorem 40, L bix)^l. If x=a L bix)=k>l, then there are two vicinities of k, V]ik) contained x=a in V2ik) and V2ik) not containing I. By Theorem 26 a Vi*ia) exists such that if x is in 7i*(a), bix) is in V^ik). Further- more, by the definition of bix), if Xi is an arbitrary value of x on Fi*(a), then there is a value of x in a Xj such that fix) is in Vik). Hence k would be a value approached by fix) con- trary to the hypothesis k>l. § 6. Bounds of Indetennination. Oscillation. It is a corollary of Theorem 43 that in the approach to any point a from the right or from the left the least upper bound and the greatest lower bounds of the values approached by fix) are themselves values approached by fix). The four num- bers thus indicated may be denoted by fia+0) = L fix) = L fix), 1=0+0 lia 84 INFINITESIMAL ANALYSIS. the least upper bound of the values approached from the right: /(a-0)=L/(x)=L/(x), x=a — x—a the least upper bound of the values approached from the left: j{a+0)=L f(x)=L fix), x'-a+O <— the greatest lower bound of the values approached from the right : /(o-0)=L/(x)=L/(x), lio- — > x±a the greatest lower bound of the values approached from the left. If all four of these values coincide, there is only one value approached and L fix) exists. If /(a+0) and fia+0) coincide, x=a ■ this value is denoted by /(a+0) and is the same as L fix). x>a x±a Similarly if /(o-O) and /(a-0 ) coincide, their common value, L fix), is denoted by /(a - 0) . The larger of /(o+O) and /(a-0) x<a is denoted by L f{x), and is called the upper limit of f{x) as X approaches a. Similarly L f(x), the lower limit of /(x), is x=a the smaller of /(a+0 ) and fja-O ). L fix) and L fix) are called the bounds of indetermination of fix) at x = a (Unbe- stimmtheitsgrenzen) . See the Encyclopadie der mathematischen Wissenschaften, II 41. In order that a function shall be continuous at a point a it is necessary and sufficient that /(a)=/(a+0)= /(a+0 )=/(o-0)= /(a+0) . ... (a) The difference between the greatest and the least of these THEORY OF LIMITS. 85 values is called the oscillc.tion of the function at the point a. It is denoted by 0(,/(x), and according to the theorem above is equivalent to the lower bound of all values of 0/(x), where 0/(3;) =Bf{x)-Bf{x) for a segment V{a). 0^(1) is used for the oscillation of fix) on the segment a b. It is sometimes also used for the oscillation of }{x) on the interval a b. The word oscillation may also be applied to the difference between the upper and lower bounds of the function on a V*{a). Denote this by Ov*(a)fi^)- The lower bound of these values may be denoted by Oa*f{x) and is the difference between the greatest and the least of the four values /(a+0), /(o-O), f(a + ), /(g-O )- The reader wiU find it a useful exercise to construct exam- ples and to enumerate the different ways in which a func- tion may be discontinuous, according as /(o+O) or /(a— 0) exist or do not exist, and according as /(o) does or does not coincide with any of the values approached by fix). (Compare the reference to the E. d. m. W. given above.) The principal classification used is into discontinuities of the first kind, where /(a+0) and /(a-0) both exist, and discontinuities of the second kind, where not both /(a+0) and /(a-0) exist. Theorem 44. If a is a limit point of [x], then a necessary and sufficient condition that &2 and bi shall be the upper and lower bounds of indetermination of fix), as x=a, is that for every set of four numbers Oi, a2, Ci, C2, such that t ai<6i<Ci<C2<62<ffl2, there exists a V*ia) such thai for every x on V*ia) ai</(x)<a2, and for som£ x', x" on V*ia) fix')>C2 and fix")<Cy. t If 6, = - 00, o, = 6, replaces a,<bu If 6a = + =0 , Oj = b, replaces 6, < Oj. 86 INFINITESIMAL ANALYSIS. Proof. — I. The condition is necessary. It is to be proved^ that if 62 and 61 are the upper and lower bounds of indeter- mination of f{x), as x = a on [x], then for every four numbers ai<bi<Ci<C2<b2<a2 there exists a V*{a) such that: — (1) For all values of x on V*{a), a^<f{x)<a2. If this con- clusion does not follow, then for a particular pair of numbers Ci, a2, there are values of f{x) greater than 02 or less than Gj for X on any V*{a), and by Theorems 24 and 40 there is at least one value approached greater than 62 or less than 61. This would contradict the hypothesis, and there is therefore a V*{a) such that for all values of x on F*(a), ai<f{x) <a2. (2) For some x', x" on F*(o), f{x')>C2 and /(x")<Ci. If this conclusion should not follow, then for some V*{a) there would be no x' such that f(x') >C2, or no x" such that f{x") <Cj, and therefore bi and 62 could not both be values approached. II. The condition is sufficient. It is to be proved that 62 and 61 are the upper and lower bounds of the values approached. If the condition is satisfied, then for every four numbers O], 02, Ci, C2, such that ai<bi<Ci<C2<b2<a2 there is a V*(a) such that for all j's on V*{a) ai</(x)<a2, and for some x', x", f{x')>C2 and f{x")<Ci. By Theorem 24 there are values ap- proached, and hence we need only to show that 62 is the least upper and 61 the greatest lower bound of the values approached. Suppose some B>b2 is the least upper bound of the values approached; 02 may then be so chosen that b2<a2<B, so that by hypothesis for x on V*(,a) B cannot be a v^lue approached. Again, suppose B<62 to be the least upper bound; c may then be chosen so that B<C2, and hence for some value x' on each F*(o), /(x')<C2. By the set of values f{x') there is at least one value approached. This value is greater than C2>B. Therefore B cannot be the least upper bound. Since the least upper bound may not be either less than 62 or greater than 62, it must be equal to 62- A similar argument will prove bi to be the greatest lower bound of the values approached. CHAPTER V. CONTINUOUS FUNCTIONS. § I. Contintiity at a Point. The notion of continuous functions will in this chapter, as in the definition on page 61, be confined to single-valued func- tions. It has been shown in Theorem 34 that if /i(i) and fzix) are continuous at a point x=a, then /lU)±/2(l), /l(x)-/2(x), /i(x)//2(x), (/2(X)?^0) are also continuous at this point. Corollary 1 of Theorem 39 states that a continuous function of a continuous function is continuous. The definition of continuity at a;=a, namely, Lf{x)=f{a), x-a is by Theorem 26 equivalent to the following proposition : For every £>0 </icre exists a S,>0 such that if \x-a\<S„ then \fix)-j{a)\<e. It should be noted that the restriction Xt^u which appears in the general form of Theorem 26 is of no significance here, since for x=a, |/(i)— /(a)|=0<£. In other words, we may deal with vicinities of the type V{a) instead of V*{a). The difference of the least upper and the greatest lower I — I bound of a function on an interval a b has been called in Chapter IV, page 85, the oscillation of /(i) on that interval, and denoted by 0*(x). The definition of continuity and Theorem 27, Chapter III, give the following necessary and suflScient condition for the continuity of a function /(i) at the 87 88 INFINITESIMAL ANALYSIS. point x=a: For every e>0 there exists a d,>0 such that if \xi-a\<d., and |x2-a|<^„ then \f{xi)-f{x2)\<K- This means that for all values of ii and xz on the segment {a — d,) (a + 5,) B\fix,)-fix2)\<^<e, and this means Bfix) —Bf{x) < e, or 0:lt'Jix)<s. Then we have Theorem 45. If fix) is continvaus for x=a, then for every e > there exists a V,{o-) such that on V,(o) the oscillation of f{x) is less than e. Theorem 46. // f{x) is continvaus at a point x=a and if f(a) is positive, then there is a neighborhood of x=a upon which the function is positive. Proof. — If there were values of x, [x'] within every neighbor- hood of x=a for which the function is equal to or less than zero, then by Theorem 24 there would be a value approached by /(a/) as a/ approaches a on the set [a/]. That is, by Theorem 40, there would be a negative or zero value approached by f{x), which would contradict the hypothesis. § 2. Continuity of a Function on an Interval. Definition. — ^A function is said to be continuous on an in- terval a 6 if it is continuous at every point on the interval. Theorem 47. If f{x) is cordinujous on a finite interval a b, then for every e>0,a b can be divided into a finite number of equal intervals upon each of which the oscillation of f{x) is less than e.f t The importance of this theorem in proving the properties of continu- ous functions seems first to have been recognized by Goursat. See his Coura d' Analyse, Vol. 1, page 161. CONTINUOUS FUNCTIONS. 89 Proof. — By Theorem 45 there is about every point oi a b a. segment a upon which the oscillation is less than e. This set of segments [o] covers a b, and by Theorem 11 a 6 can be divided into a finite number of equal intervals each of which is interior to a <i; this gives the conclusion of our theorem. Theorem 48. {Unijorm continuity.) If a function is con- I — I tmuous on a finite interval a b, then for every £ > there exists a d,>0 such that for any two values of x, Xi, and X2, on a b where \Xi-X2\<d,, |/(Xi)-/(X2)|<£. Proof. — ^This theorem may be inferred in an obvious way from the preceding theorem, or it may be proved directly as follows : By Theorem 27, for every e there exists a neighborhood V^(x') of every a/ of a b such that if Xi and X2 are on V (x'), then |/(a;i)-/(x2)|<£. The V'{xfys constitute a set of seg- I— I ments which cover a b. Hence, by Theorem 12, there is a, d, such that if \xi—X2\>S„ Xi and X2 are on the same F(x') and consequently |/(xi)— /(x2)l<e. The uniform continuity theorem is due to E. Heine.! The proof given by him is essentially that given above. In 1873 LiJROTH J gave another proof of the theorem which is based on the following definition of continuity : A single-valued function is continuous at a point x=a' if for every positive e there exists a d, such that for every Xi and X2 on the interval a-d, a + S„ |/(a;i)-/(x2)|<« (Theorem 45). By Theorem 42 there exists a greatest d for a given point and for a given e. Denote this by J,(x). If the function is con- I — I tinuous at every point of a b, then for every £ there will be a value of J,{x) for every point of the interval, i.e., d,{x), for any particular e, will be a single-valued function of x. t E. Heine: Die Elemente der Functionenlehre, Crelle, Vol. 74 (1872), p. 188. 1 Luroth: Bemerkung iiber Gleichmdssige Stetigkeit, Mathematische Anna- len, Vol. 6, p. 319. 90 INFINITESIMAL ANALYSIS. The essential part of Luroth's proof consists in establishing the following fact: If f(x) is continuous at every point of its interval, then for any particular value of e the function i/x) is also a continuous function of x. From this it follows by Theorem 50 that the function J,{x) will actually reach its greatest lower bound, that is, will have a minimum value; and this minimum value, like all other values of d„ will be positive.f This minimum value of J (x) on the interval under consideration will be effective as a <?, independent of x. The property of a continuous function exhibited above is called uniform continuity, and Theorem 48 may be briefly stated in the form: Every function amtimwus on an interval is uniformly continuous on that interval.t This theorem is used, for example, in proving the integrability of continuous functions. See page 157. I — I . Theorem 49. // a function is continuous on an interval a b, it is bounded on that interval. Proof. — By Theorem 46 the interval a b can be divided into a finite number of intervals, such that the oscillation on each interval is less than a given positive number s. If the number of intervals is n, then the oscillation on the interval a 6 is less than ne. Since the function is defined at all points of the interval, its value being f{xi) at some point Xi, it follows that I — I every value of f(x) on a 6 is less than f{xi) +ne and greater than /(xi)-n£; which proves the theorem. Theorem 50. // a function f{x) is continuous on an interval * t It is interesting to note that this proof will not hold if the condition of Theorem 26 is used as a definition of continuity. On this point see N. J. Lennes: The Annals of Mathematics, second series, Vol. 6, p. 86. Jit should be noticed that this theorem does not hold if "segment" is substituted for "interval," as is shown by the function — on the segment 1, which is continuous but not uniformly continuous. The function is defined and continuous for every value of x on this segment, but not for every value of X on the interval 1. CONTINUOUS FUNCTIONS. 91 I — I a b, then the function assumes as values its least upper and its greatest lovoer hound. Proof.— By the preceding theorem the function is bounded and hence the least upper and greatest lower bounds are finite. By Theorem 19 there is a point k on the interval a b such that the least upper bound of the function on every neighborhood of x=kis the same as the least upper bound on the interval a b. Denote the least upper bound of f{x) on a 6 by B. It follows from Theorem 43 that 5 is a value approached by fix) as x approaches k. But since L f{x)=f{k), the function being con- tinuous at x=-k, we have that f{k) = B. In the same manner we can prove that the function reaches its greatest lower bound. Corollary. — If A; is a value not assumed by a continuous , ■ . I — I function on an interval a b, then f{x) -k or k-f{x) is a con- tinuous function of x and assumes its least upper and greatest lower bounds. That is, there is a definite number i which is the least difference between k and the set of values of /(x) 'on the interval a b. I — I Theorem 51. If a function is continuous on an interval a b, then the function takes on all values between its least upper and its greatest lower bound. Proof. — If there is a value k between these bounds which is not assumed by a continuous function f{x), then by the corollary of the preceding theorem there is a value i such that no values of fix) are between k—J and k+J. With e less than J divide the interval a b into subintervals according to Theorem 47, such that the oscillation on every interval is less than £. No interval of this set can contain values of fix) both greater and less than k, and no two consecutive intervals can contain such values. Suppose the values of fix) on the first interval of this set are all greater than k, then the same is 92 INFINITESIMAL ANALYSIS. true of the second interval of the set, and so on. Hence it I — I follows that all values of f{x) on a b are either greater than or less than k, which is contrary to the hypothesis that k Ues be- tween the least upper and the greatest lower bounds of the function on a b. Hence the hypothesis that /(x) does not assume the value k is untenable. By the aid of Theorem 51 we are enabled to prove the fol- lowing : Theorem 51a. // /i(x) is continuous at every point of an inier- ' — I val a' b' except at a certain point a, and if Lfi{x) = + cc and L/2(x)= — 00, then for every b, finite or +00 or — 00 , there exist two sequences of points, [xj] and [x/] (i=0, 1, 2, . . . ), each sequence having a as a limit point, such that L i/ife)+/2(x/)!=6. Proof. — Let [x/] be any sequence whatever on a' b' having a ... . I — I as a limit point, and let Xq be an arbitrary point of a' V. Since /i(x) assumes all values between /i(xo) and +00, and since ^ f2{x) = — oo, it follows, in case b is finite, that for every i greater than some fixed value there exists an Xi such that fl{Xi)+f2ix/)=b. In case 6 = -I- « , Xj is chosen so that fl{Xi)+f2iXi')>i. Corollary. — Whether /j(x) and /2(x) are continuous or not, if L /i(x)=4-oo and L /2(x)=-oo, there exists a pair of CONTINUOUS FUNCTIONS. 93 sequences [xj and [x/] such that L {/l(Xi)+/2(x/)| »=« is + 00 or — 00 . Theorem 52. // y is a function, f{x), of x, monotonic and con- I — I tinuous on an interval a b, then x=f~^{y) is a function of y which is monotonic and continuous on the intenal f{a) f{b). Proof. — By Theorem 20 the function f~'^{y) is monotonic and has as upper and lower bounds a and h. By Theorems 50 and 51 the function is defined for every value of y between and including /(a) and f{b) and for no other values. We prove the I — I function contmuous on the interval /(a) f{b) by showing that it is continuous at any point y=yi on this interval. As y I- — I approaches j/i on the interval /(a) j/i, f~^(y) approaches a definite limit g by Theorem 25, and by Theorem 40 a<glf-^{yi)±b. If g<f~^{yi), then for values of x on the interval g /(j/j) there is no corresponding value of y, contrary to the hypothesis that f{x) I — 1 is defined at every point of the interval a b. Hence g=f~'^{yi), and by similar reasoning we show that f~^{y) approaches /~H2/i) I — —*] as y approaches 2/1 on the interval, 2/1 / ^{b). Theorem 53. // f{x) is single-valued and continuous with A, B I — I as lower and upper bounds, on an interval a b and has a single- I — I . I— I valued inverse on the interval, A B then f{x) u monotonic on a b. Proof. — If f{x) is not monotonic, then there must be three values of x, Xi<X2<X3, such that either f{xi)^f(x2)'^f{xz) or /(Xi)^/(X2)/^(X3). In either case, if one of the equality signs holds, the hypothesis tnat /(x) has a single-valued inverse is contradicted. If there 94 INFINITESIMAL ANALYSIS. are no equality signs, it follows by Theorem 51 that there are two values of x, Xt and x^, such that Xi<X4<X2<3;5<2:3, and /(X4)=/(X5), in contradiction with the hypothesis that f{x) has a single-valued inverse. Corollary.— li f{x) is single-valued, continuous, and has a single-valued inverse on an interval a b, then the inverse func- I — I tion is monotonic on A B. § 3. Functions Continuous on an Everywhere Dense Set. Theorem 54. // the functions fiix) and /2(x) are continuous on the interval a b, and if fi{x)=f2{x) on a set everywhere dense, then /i(x) =/2(2) on the whole interval.^ I — I Proof. — Let [a/] be the set everywhere dense on a 6 for which, by hypothesis, /i(x) =fi{x). Let x" be any point of the interval not of the set [a/]. By hypothesis x" is a limit point of the set [x'], and further /i(x) and /2(x) are continuous at x=x". Hence L /i(x) =/i(x") and L /2(x) =/2(x"). X=l" But by Theorem 41 L fi{x')=L fi (x) , x'=i" lil" and by Theorem 41 L f2{xf) =L fzix). x'il" x=x" Therefore /i(x") =/2(x"). t I.e., if a function f(x), continuous on an interval a b, is known on an everywhere dense set on that interval, it is known for every point on that interval. CONTINUOUS FUNCTIONS. 95 I — I Definition. — On an interval o 6 a function /(x') is uniformly continuous over a set [x'] if for every £>0 there exists a 5,>0 such that for any two values of a/, Xi, and X2' an a b, for which \xi' -X2'\<d„ \}{x,').-f{x2')\<e. Theorem 55. // o function f{x') is defined on a set everywhere I— I dense on the interval a b and is uniformly continuous over that set, then there exists one and only one function f{x) defined on I — I the full interval a b such that: (1) /(i) is identical with f{xf) where fix') is defined. 1 — I (2) f{x) is continuous on the interval a b. I — I Proof. — Let x" be any point on the interval a b, but not of the set [a/]. We first prove that L/(x') x' £1" exists and is finite. By the definition of uniform continuity, for every e there exists a d, such that for any two values of x', Xi, and x/, where |xi'-i2'|<5., |/(xi') -/(a;2')l<^- Hence we have for every pair of values Xi' and Xz where |xi'-x"|<y and 1x2' -x"| <Y that |/(xiO -/(xz') | < £. By Theorem 23 this is a sufficient condition that L/(x') x-^x" shall exist and be finite. Let fix) denote a function identical with fixf) on the set [x'] and equal to Lfix') at all points x". This function is defined upon the continuum, 96 INFINITESIMAL ANALYSIS. since all points x" on a b are limit points of the set [a/]. Hence the function has the property that L /(x')=/(x) for every x Xi -X of a b. We next prove that f{x) is continuous at every point on the I — I interval a b, in other words that f{x) cannot approach a value b different from /(zi) as x approaches Xi. We already know that /(z) approaches f{xi) on the set [x']. If 6 is another value approached, then for every positive e and d there is an x^ such that \X.,-Xi\<d, \f{X.,)-b\<€ (1) Since f{xa) =L f{x') we have that for every e>0 there exists a d,>0 such that for every x' for which \x' -x^| <5„ \fix')-f{x.,)\<e (2) From (1) and (2) we have \m-b\<2e (3) Since the d of (1) is any positive number, there is an x^ on every neighborhood of xj, and hence by (2) and (3) an a/ on every neighborhood of Xi such that |/(x') — ?>| <2£, e being arbi- trary and b a constant different from /(xi"). But this is con- trary to the fact proved above, that L j{x') exists and is equal to /(xi). Hence the function is continuous at every point of the I— I interval a b. The uniqueness of the function follows directly from Theorem 54. , This theorem can be applied, for example, to give an ele- gant definition of the exponential function (see Chap. III). We first show that the function a" is uniformly continuous on the set of all rational values between xi and X2, and then define CONTINUOUS FUNCTIONS. 97 a* on the continuum as that continuous function which coin- — TO cides with a" for the rational values — . The properties of the function then follow very easily. It will be an excellent exer- cise for the reader to carry out this development in detail. § 4. The Exponential Function. Consider the function defined by the infinite series x^ x^ x^ Applying the ratio test for the convergence of infinite series we have .71! ■ (n-1)! n If n' is a fixed integer larger than x, this ratio is always less than — < 1 The series (1) therefore converges absolutely for every n' value of X, and we may denote its sura by e(x). From Chap. I, page 17, we have that Theorem 56. .iMT- where [n] is the set of att positive irUegers, exists and is equal to e(x) for all values of x. 98 INFINITESIMAL ANALYSIS. Proof.— Let E„(x) = 2' ri fc=oA;! (where 0! = 1). Then, since \ n) " ■^(n-l)!'n"^(n-2)!-2!W "^•"'^nlW ' it follows that (x\ "I " / 1 t? ^ \ 1+-) = I (t-.-t ,,,• , J x* < " /I to(w-I) ■■■ (n-A: + l) )m < -^ ri~~fc F • A;!n* Now, since n*-(n-A; + l)* = (A-l){n*-i+n*-2.(n-A + l)+... + (n-A; + l)*-M<(i-l)A;-n*-i, it follows that ^„(x)-(l+3" < 2' .^•e{\x\) ~k=.2ik-2)l-n^ n For a fixed value of x, therefore, we have (l+|)"=E„(x) + ei(»), where ei(n) is an infinitesimal as n= oo. At the same time e{x)=E„{x) + £2{n), where e2{n) is an infinitesimal as n= w. Hence L (l+-Y=e(x). CONTINUOUS FUNCTIONS. 99 Theorem 57. L (iH — ) , where [z] is the set of all real numbers, exists and is equal to e{x). Proof. — If z is any number greater than 1, let nz be the integer such that ni^z<nz + l. Hence,it«>0, l+£i l+|>l + j^j ■ . . . . (1) Henc (i+|)"""Mi+j)'>('+S:Ti)""' ' ' <^' « (-3(-3""=(-?)'>(-i:Vi)-*"-V- (3) Since L (l+f ) =1, and L (l+r4l)=l. and L (l+f) =e(x), and L (l+Z-Tj) =e(a;;, the inequaUty (3), together with Corollary 3, Theorem 40, leads to the result: L (l+4)'=e(x). The argument is similar if i<0. Corollary. ^L^(l+-) =e{x), ■where [z] is any set of numbers with limit point + oo. Theorem 58. The function e{x) is the same as e^ where 1 1 , 100 INFINITESIMAL ANALYSIS. Proof. — By the continuity of 3^ as a function of 2 (see Corol- lary 2 of Theorem 39), it follows that, since L (1+-)" =e, L (1+-) =e'. where 2=na;. Hence by Theorem 39 &=L (1+-)' and by the corollary of Theorem 57 the latter expression is equal to e(x) , Hence we have ex = i+a;+|+|^+ (1) (1) is frequently used as the definition of e*, a' being defined as e^logeO. CHAPTER VI. INFINITESIMALS AND INFINITES. § I. The Order of a Function at a Point. An infinitesimal has been defined (page 75) as a function f{x) such that L fix) =0. A function which is unbounded in every vicinity of a; = a is said to have an infinity at a, to be or become infinite at x = a, or to have an infinite singularity at x = a.t The recipro- cal of an infinitesimal at x=a is infinite at this point. A function may be infinite at a point in a variety of ways : (a) It may be monotonic and approach +00 or - 00 as x=a; for example, - as a; approaches zero from the positive side. (5) It may oscillate on every neighborhood of x=a and still approach + 00 or — 00 as a unique limit; for example, • 1 ^ sm--l-2 X as X approaches zero. t It is perfectly compatible with the.se statements to say that while fix) has an infinite singularity at x=a, /{o)=0 or any other finite number. For example, a function which is — for all values of x except x = is left undefined for i=0 and hence at this point the function may be defined as zero or any other number. This function illustrates very well how a function which has a finite value at every point may nevertheless have infinite smgularities. 101 102 INFINITESIMAL ANALYSIS. (c) It may approach any set of real numbers or the set of all real numbers; an example of the latter is . 1 sm- X as X approaches zero. See Fig. 13, page 64. (d) + 00 and — » may both be approached while no other number is approached; for example, - as a; approaches zero from both sides. Definition of Order. — If /(x) and 0(x) are two functions such that in some neighborhood V*{a) neither of them changes sign or is zero, and if where k is finite and not zero, then f{x) and <f>{x) are said to be oi the same order aX- x= a. If x=a<{>ix) ' then f{x) is said to be infinitesimal with respect to 4>{x), and <fi{x) is said to be infinite with respect to f(x). If L -rr^ = + 00 or — 00 then, by Theorem 37, <f>ix) is infinitesimal with respect to f(x), and fix) infinite with respect to <f){x). If f{x) and <p{x) are both infinitesimal at x=a, and f(x) is infinitesimal with respect to <l){x), then fix) is infinitesimal of a higher order than <^(x), and ^ix) of lower order than fix). If ^(x) and fix) are both infinite at x=o, and fix) is infinite with respect to ^(x), then /(x) is INFINITESIMALS AND INFINITES 103 infinite of higher order than <i>{x), and 0(x) is infinite of lower order than /(x).t The independent variable x is usually said to be an infini- tesimal of the first order as x approaches zero, x^ of the second order, etc. Any constant j^O is said to be infinite of zero order, - is of the first order, ^ of the second order, etc. This usage, however, is best confined to analytic functions. In the general case there are no two infinitesimals of consecutive order. Evi- dently there are as many different orders of infinitesimals be- tween X and a;2 as there are numbers between 1 and 2; i.e., xi+* is of higher order than x for every positive value of k. c,. -r /l(x) 1 , ^ foix) oince L j-T-T^T whenever L '-rT~=k, we have x-aj2\X) K x=a hW Theorem 59. If /i(x) is of the same order as fzix), then fzix) is of the same order as f\{x). Theorem 60. The function cf{x) is of the same order as f{x), c being any constant not zero. Proof.— By Theorem 34, L ^^=c. Theorem 61. If fiix) is of the same order as fiix), and /2(x) is of the savfie order as fsix), then fiix) and fsix) are of the same order. t This definition of order is by no means as general as it might possibly be made. The restriction to functions which are not zero and do not change sign may be partly removed. The existence of x=a't>(.x) is dispensed with for some cases in § 4 on Rank of Infinitesimals and In- finites. For an account of still further generalizations (due mainly to Cauchy) see E. Borel, Series a Termes Positifs, Chapters III and IV, Paris, 1902. An excellent treatment of the material of this section together with extensions of the concept of order of infinity is due to E. Borlotti, CaJr colo degli Infinitesimi, Modena, 1905 (62 pages). 104 INFINITESIMAL ANALYSIS. Proof. — By hypothesis L 7^,— - = fci and L r-r-r=k2. By Theorem 34, ^L ^-^ ;£/^=,£ WV (By definition, /2(a;) 5^0 and jz{x) y^O for some neighborhood of x=a.) Hence 7- A^^) 7 7 x=al3W Theorem 62, 7/ /i(x) and /2(x) are infinitesimal (infinite) and neither is zero or changes sign on some V*{a), then f\{x) ■f2{x) is infinitesimal (infinite) of a higher order than either. Proof. L^-^fy^^=L/i(x)=0. (±00.) xia /2W xia Theorem 63. If fi(x), . . . , fn(x) have the same sign on some V*(a) and if f2(x), . . . , fn(x) are infinitesimal (infinite) of the same or higher (lower) order than fi(x), then fl(x)+f2(x)+f3(x)+...+fn(x) is of the same order as fi(x), and if /2(x), faix), . . . , f„(x) are of higher (lower) order than fi(x), then fi(x)±f2(x)±f3(x)±. . .+fn(x) is of the same order as /i (x) . Proof. — ^We are to show that J fl(x)+f2(x)+...+f„(x) i=a fl(x) By hypothesis, xi fAx) -"" xta fl(x) -"" •'•' xia M^^^"' A T A(^) 1 INFINITESIMALS AND INFINITES. 105 Hence, by Theorem 30, r [hix) f2ix) hix) Ux)\ since all the ^'s are positive or zero. Similarly, under the second hypothesis. ^ Mx)±f2(x)±...±Ux) ^ j^ /iJ^ + M^li +/"W ] \h(x) hix) ••• /i(x)i = 1+0 + . ..+0 = 1. Theorem 64. — // /3(x) and fi{x) are infinitesimals with re- spect to fi{x) and /aCx), then J \h{x)+fz{x)\-\h{x)+j^{x)] ^ , J l/i(a:)+/3(x}-l/2(x)+/4(x)| Froot. L ,/,.,, , ^ ^ h(x) -hix) +fx(x) -Uix) +/3(X) ■f2{x) +f3ix) -f.jx ) xj,a h(x)-f2{x) J /l(x)-/2fa) , J /1(X)-/4 (X) J hix)-f2(.x) f3{x)-U(x) xLhix) •/2(X) xl'a hix) ■f2{x) JlafM ■ h{x) \Lh{x) -^ix) ^• § 2. The Limit of a Quotient. Theorem 65. If as x = a, £i(x) is an infinitesimal vnth respect to /i(x) and szix) with respect to fzix), then the valv£s approached by /l(x) + £l(x) ^^^ M£) /2(X)+«2(X) ^^ /2(X) •as X approaches a are identical. 106 INFINITESIMAL ANALYSIS. Proof.— This follows from the identity ^i(x)\ /l(x)+£i(x) _ /i(x) f2{x)+e2{x) fiix)' /i ,i2_(i) 1 + f2{x)' Since tt4 and 4^,-^ are infinitesimal. flix) }2{X) Corollary. — If /i(x) and /2(x) are infinite at a;=a, then fiix)+c /i(x) f,{x)+d ^""^ /2(X) approach the same values. Theorem 66. // L ^,= L ^,^k, and if L^,=l • ^ V ,. i. T hi. x)+h{x) J hix) IS finite, then lc= L -,—, ^ , , , ^ = Li , , . , ' ' x^a^\{x)-V^2{x) xUaA-^^^y provided l^ -1 if k is finite, and provided l>0 if k is infinite. fi{x)+f2{x) f2{x) fi(x)<j>2{x)-f2{x)Mx) Proof. 4>iix)+Mx) Mx) Mx){<f>i{x)+Mx)) ' /l(x)+/2(x) _ Mx) //i(x) /2(X)\ 7 1 \ 4>l{x)+(j>2{x) 4>2{X) \(/)i(x) (f>2{x)l "I ^2{X) ). In case k is finite, the second term of the right-hand member is evidently infinitesimal if Ij^—l and the theorem is proved. In the case where k is infinite we write the above identity in the following form: /1(X)+/2(X) /l(x) 1__.^,/2(X) 1 4>i{x)-\-<f>2(x) 0i(x) ^2(3:) (f>2{x) 9!>i(x) ' «^i.(x) %2(a;) INFINITESIMALS AND INFINITES. 107 Both terms of the second member approach + oo or both - oo if Z>0. Corollary. ~li <f>i{x) and 962(3;) are both positive for some VHa), and if A= L ilM= L ^4^, then L i#tM^ =k whenever k is finite. If k is infinite, the condition must be added that , , - has a finite upper and a non-zero lower bound. Theorem 67. // /i(x) and /2(x) are both infinitesimals asx = a, then a necessary and sufficient condition that L , , - =k {k finite and not zero) x = a /2W is that in the equation fi(x)=k-f2{x) + s(x), e{x) is an infini- tesimal of higher order than /i(x) or fzix). Proof. — {lyThe condition is necessary. — Since L r~-( =/fc, X4a/2(X) }2{X) '' + '^^'^^' or /i(x)=/2(2;)-A;+/2(x)-£'(a;), where L £'(x)=0 (Theorem 31). x~a By Theorems 60 and 61, fi(x) and /2(x) -k are of the same order, since kf^O, while by Theorem 62 s'{x)-f2{x) is of higher order than either /i(x) or f2(x). Hence the function s{x) = s'(x) •/2(x) is infinitesimal. (2) The condition is sufficient. — By hypothesis /i(x) = f2(,x)-k + £(x), where /i(x) and /2(x) are of the same order as £(t) x=a, while e(x) is of higher order than these. Let e'{x) =ryT, /2W / (x) which by hypothesis is an infinitesimal. We then have j-j— t / \ = k + e'{x). Hence, by Theorem 31, L j-j-r = k. 108 INFINITESIMAL ANALYSIS. § 3. Indeterminate Forms.t a Lemma. — // j- and -y are any two fractions such that b and d are both positive or both negative, then the value of a + c b+d I — I ,. , . . a c lies on the interval j- -7. Proof. — Suppose b and d both positive and a ^a+c b^b+d' then db+ad^ab+ be. .'. ad^ be; .'. cd+ad^ cd+bc; a+c ^c • ■ 6+d = d* The other cases follow similarly. Theorem 68. // /(x) and (f>{x), defined on some F(+qo), are 'both infinitesimal as x approaches + 00 , and if for some positive number h, (f>{x+h) is always less than <f>{x) and , f{x+h)-f{x) ^^ then „4>{x+h)—4>{x) exists and is equal to k.X f The theorems of this section are to be used in § 6 of Chap. VII. % This and the following theorem are due to O. Stolz, who generalized them from the special oases (stated in our corollaries) due to Cauchy. See INFINITESIMALS AND INFINITES. 109 Proof.— Let Vi(k) and Vzik) be a pair of vicinities of k such that Vzik) is entirely within T^(^-). By hypothesis there exists an h and an A'o such that if j > A'2, f{x + h)-fix) (l>{x+h)-i>{x) W is in V2{k). Since this is true for every x>X2, f{x + 2h)-fix + h) <i>ix+2h)-<j,{x+h) (2) is also in V2{k). From this it follows by means of the lemma fV,of f(x+2h)-fix) ^^^^ <l>{x+2h)-<pixy (3) whose value is between the values of (1) and (2), is also in V2{k). By repeating this argimient we have that for every integral value of n, and for every x>X2, jix+nh)-f{x) (j)ix+nh)—<p{x) is in F2(*)- By Theorem 65, for any x ^ fjx+m-fix) _ /(i) n=<x><f>{x+nh)-^{x) <j>{xy Hence for every a; and for every e there exists a value of n, Nx„ such that if n> Nxi, f{x+nh)-fix) fix) <j>{x+nh)—(j)(x) <f>{x) <£. Taking e less than the distance between the nearest end-points fix) cf Viik) and V2ik) it is plain that for every i>A'2, ttt is (pyX) Stolz und Gmeiner, Functionentheorie, Vol. 1, p. 31. See also the referenca to BoBTOLOTn given on page 103. 110 INFINITESIMAL ANALYSIS. on Vi{k), which, according to Theorem 26, proves that Corollary. — If [n] is the set of all positive integers and ^(n + 1) < <pin) and f(n) and ^(n) are both infinitesimal as w = qo , then if /(n + l)-/(n) _ it follows that L -rr^ exists and is equal to k. Theorem 69. If }{x) is bounded on every finite interval of a certain y( + oo), and if ^{x) is monotonic on the same F( + oo) and L ^(x) = +00 , and if for some positive number h fix+h)-f{x) ,^^<j>{x+h)-<}>(x) "' then L ~rl exists and is equal to k. Proof. — By hypothesis, for every pair of vicinities Vi{k) and y2{k), V2(k) entirely within Vi(k), there exists an X2 such that if x>X2, then f{x+h)-f{x) 4{x+h)-<t>{x) is in Viik). From this it follows as in the last theorem that f{ x+nh)-f{x) <j>(x+nh) —<f>{x) is in Vzik). Now make use of the identity fjx+nh) _ f{x+nh)-f{x) fix) 4>{x+nh) (j>{x+nh) ^{x+nh) INFINITESIMALS AND INFINITES. Ill f{x+nh)-fix) / <f>{x) \ fix) '4>{x+nh)-<j>{x)\ <j>ix+nh)/'^<f>{x+nhy ' ' ^^ Let [a/] be the set of all points on the interval X2 X2+h, and for this interval let A2 be an upper bound of \f{x^) | and B2 an upper bound of <j>{x'). Then cl>{x') _ B2 4>{xf +nh) ~ ''^^' "^ ^ <j>{X2+nh) and JL/!./ , An = «2(a/, n) < — - — <^(i' +n;i) ~ ''^ ' '"^ ^ <]>{X2 +nhy Hence for every £ there exists a value of n, N^y, such that if n>N£y^ £1(2/, n)<e and £2(2^, «)<« .... (2) I — I independently of xf so long as a/ is on X2 X2+h. There are then three cases to discuss: (1) A finite. (2) k= + <». (3) 4= -00. (1) k finite. By the preceding argument, for x>Z2, f{x+nh)-f{x) <j>{x+nh)-<j){x) is in V2(k), and hence \fixf+nh)-f{xf)\ ^^, . where ey, is the length of the mterval FaCA) and K the absolute value of k. Then, in view of (1), fi^+r^ _ fjif +nh) -fix') 4>ix!+'nh) <l>ix'+nh)-4>ixf) <iK+ev^€iix/,n) + e2ix',n). 112 INFINITESIMAL ANALYSIS. Now take £v smaller in absolute value than the length of the interval between the closer end-points of Vi(k) and V2ik). By (2) there exists a value of n, N^y, such that if n>N£y, -'i(^»<2(Ztt;;:) and e2(x',n)<-^ for all values of 2/ on X2 X2 +h. Hence for n> AT J J, f(x'+nh) fix' + nh) -f{x') 4){x'+nh) ^{x'+nh)-4>{x') «7 , «v <^^+^^.)2(:^TT7;+T^'^ and since for a;> Z2 +Neyh there is an n> iVjj, and an a/ between X2 and X2+h such that 3/+nh=x, it follows that if a;> X2 +Ney, ]Kx) f(x'+nh)-fix') l |^(x) 4>(.^+nh)-<f>ix)\'^^^' fix) and therefore, -j-p-!- is on Fi(A). This means, according to Theorem 26, that (2) A=+oo. If the numbers wii and ma are the lower end pomts of 7i(A;) and Y^ih), then <^(x'+nA)-,^(x')>"'2 forx'>Z2. INFINITESIMALS AND INFINITES. 113 If ev is then chosen less than m2-mi, there will exist a value of A'^g^ such that for all values of n>Ngy independently of x' so long as a/ is in I X2 X2 + h. Then, in view of (1), f{x'+nh) ii ^v \ ey ev 1 1 \ Since there is no loss of generality if m2> +1, this proves that for X >X2 +Nsyn, fix) >m2 — £v>mi, <P{x) and hence -rrr is on VAk). 4>{x) (3) A; = — 00 is treated in an analogous manner. Corollary 1. If [n] is the set of all positive integers and if <j>{n-\-\)> ^(n) and L ^(n) = 00 , 11 = 00 . .f r /(n + l)-/(n) then If „f.^(n + l)-^(n)-*' . it follows that L -yt-x exists and is equal to k „-=o?>W I — I Corollary 2. If f{x) is bounded on every interval, x (x + l), and if L f{x + l)-}{x)=k, I— 00 m then L XS3 00 x exists and is equal to k. 114 INFINITESIMAL ANALYSIS. § 4. Rank of Infinitesimals and Infinites. Definition. — If on some V*{a) neither /i(a;) nor /2(x) vanishes, and fi(x) f2ix) and Ihix) are both bounded as x approaches a, f (x) then fi{x) and /2(x) are of the same rank whether L j-i—. exists or not.t The following theorem is obvious. Theorem 70. // /i(x) and /2(x) are of the same order, they are of the same rank, and if fi{x) and f2{x) are of different orders, they are not of the same rank. If fi{x) and ^(a;) are of the same rank, they may or may not be of the same order. Theorem 71. // fi{x) and fiix) are of the same rank as x approaches a, then c-fi{x) and /2(x) are of the same rank, c being any constant not zero. Proof. — By hypothesis for some positive number M, AW hence /2(X) c-fiix) <M and /2(X) hix) <M, /2(X) <M-|c| and f2{x) c-fi{x) < M Theorem 72. // /i(x) and /2(x) are of the same rank and fiix) and fz{x) are of the same rank as x approaches a, then fi{x) and fz{x) are of the same rank as x approaches a. Proof. — By hypothesis, fi{x) /2(X) <Mi and /2(X) hix) <M2 in some neighborhood of 2;= a. Therefore fiix) f2{x) /2(X) /3(X) <Mi-M2 or /i(x) fsix) <Mi-M2. 1 1 and I -(sin — 1-2) are of the same rank but not of the same order as I approaches zero. INFINITESIMALS AND INFINITES. In the same maimer 115 /2(X) hix) <Mi and fsix) f2{x) <M2, whence hix) hix) <Mi-M2. Theorem 73. // /i(x) is infinitesimal {infinite) and does not vanish on some V*{a), and if fiix) and fz{x) are infinitesimal {infinite) of the sam£ rank as x approaches a, then fi{x) ■f2{x) is 0/ higher order than /aCx), and f\{x) ■fz{x) is of higher order than }2{x). Conversely, if for every function, fi{x), infinitesimal {infi- nite) at a, fi{x) ■f2{x) is of higher order than f3{x), and fi{x) ■fz{x) is of higher order than f2{x), then f2{x) and /3(x) are of the same rank. fiix) Proof. — Since fsix) by Theorem 33 that is bounded as x approaches a, it follows ^ fi{x)-f2{x) _^ fsix) which proves the first part of the theorem. fsix) Since Ukewise /2(X) is bounded, we have that flix)-f3{x) L f2ix) Suppose that for every fiix) flix)-f2ix) =0. L 1=0 fsix) =0 and L^^^W^^O, and that /2(x) and fsix) are not of the same rank. Then, on a certam subset [x J, L r%fs{x') f2{x) same 0, or on some other subset [x"l L ^=0. Let /i(x)=(7§ on the set [x'] for which x'±al2iX ) /3W ^ M£)^Q j^j^d x-a on the other points of the continuum; x=.Js{x) ' 116 INFINITESIMAL ANALYSIS. then /i(x) is an infinitesimal as x approaches a, while for the set [x'] h{xf)-k{xf) _ Mxo U^), which contradicts the hypothesis that J h(x)-f3ix) Li T~7~\ ~^' x^a I2(X) Similarly if on a certain subset L r-r^=0, we obtain a con- t / \ tradiction by putting /i(x) =rT~:' CHAPTER VII. DERIVATIVES AND DIFFERENTIALS. § I. Definition and Illustration of Derivatives. Definition.— If the ratio Z^_^ approaches a definite limit, finite or infinite, as x approaches Xi, the derivative of f{x) at the point Xi is the limit J- fix) -/fa) _ x=xi X Xi It is implied that the function }{x) is a single-valued function Oh Fig. 14. of X. x—Xi is sometimes denoted by Jxi, and f{x)—f{xi) by J/(xi), or, if 2/=/(x), by Ayi. An obvious illustration of a derivative occurs in Cartesian geometry when the fimction is represented by a graph (Fig. 14). 117 118 INFINITESIMAL ANALYSIS. Here ^^^^ ^^^^^ is the slope of the line AB. If we sup- X — Xi pose that the line AB approaches a fixed du-ection (which in this figure would obviously be the case) as x approaches Xi, f(x) —f(xj) then L -^^ — -^"-^ will exist and will be equal to the slope of the limiting position of A B. If the point x were taken only on one side of x\, we should have two similar limiting processes. It is quite conceivable, how- ever, that limits should exist on each side, but that they should differ. That case occurs if the graph has a cusp as in Fig. 15. Fig. 15. These two cases are distinguished by the terms progressive and regressive derivatives. WTien the independent variable approaches its limit from below we speak of the progressive derivative, and when from above we speak of the regressive derivative. It follows from the definition of derivative that, except in one singular case, it exists only when both these limits exist and are equal. The exception is the case of a derivative of a function at an end-point of an interval upon which the function is defined. Obviously both the progressive and the regressive derivative cannot exist at such a point. In DERIVATIVES AND DIFFERENTIALS. 119 this case we say the derivative exists if either the progressive or the regressive derivative exists. Whether the progressive and regressive derivatives exist or not, there exist always four so-called derived numbers (which may be± w), namely, the upper and lower bounds of indeter- mination of /(x)-/(xi) X—Xi ' a? x = Xi from the right or from the' left. (Compare page 84, Chapter IV.) The derived numbers are denoted by the sjonbols D, D, D, D, analogous to the symbols on page £4. Of cotirse, in every cEise^ D^D and D^D. — > < — If we consider the curve representing the function . 1 " X at the point 2;=0, it is apparent that the Umiting position of A B does not exist, although the function is continuous at the point x=0 if defined as zero for x=0. For at every maxi- mum and minimum of the curve sin—, a; -sin = ±x, and the X X curve touches the lines x=y and x=—y. That is, • — — - — — ^ ' X—Xi approaches every value between 1 and —1 inclusive, as x approaches zero. The notion derivative is fimdamental in physics as well as in geometry. If, for instance, we consider the motion of a body, we may represent its distance from a fixed point as a function of time, /(<). At a certain instant of time h its dis- tance from the fixed point is /(ii), and at another instant ^2 it is f(t2) ; then ti—tz course indicated by the sign of the expression _ 120 INFINITESIMAL ANALYSIS. is the average velocity of the body during the interval of time ti-t2 in a direction from or toward the assumed fixed point. Whether the motion be from or toward the fixed point is of f{t l}-f(t2) . If we consider this ratio as the time interval is taken shorter and shorter, that is, as <2 approaches h, it will in ordinary physical motion approach a perfectly definite limit. This limit is spoken of as the velocity. of the body at the instant ti. Definition. — ^The derivative of a fimction y=f{,x) is denoted df(x) diV by fix) or by Di/(x) or -~^ '^^ T" /'(^) ^ ^^^° referred to as the derived Junction of /(x) . § 2. Formulas of Differentiation. Theorem 74. The derivative of a constant is zero. More precisely: If there exists a neighborhood of Xi such that for every valus of X on this neighborhood fix) =/(ii), then fixi) =0. Proof. — In the neighborhood specified =0 for every value of x. Corollary. — If f(xi) exists and if in every F*(xi) there is a value of X such that fix) =fixi), then fixi) =0. Theorem 75. When for two functions /i(x) and fzix) the derived functions fi'ix) and fz'ix) exist at xi it follows that, except in the indeterminate case 00 — 00 , (a) // fsix) =/i(x) +/2(x), then fzix) has a derivative at Xi and /3'(Xl)=/l'(Xi)+/2'(Xi). (b) If fsix) =/i(x) •f2{x), then fsix) has a derivative at Xi and fz'ix{) =/i'(xi) -/aCxi) +/i(xi) -//(xi). (c) // fzix) =-7-r\, then, provided there is a F(xi) wpon which ./2(x) 7^0, fzix) has a derivative and ,,, . //(Xl)-/2(Xl)-/l(Xi)-//(Xi) DERIVATIVES AND DIFFERENTIALS. 121 Proof. — By definition and the theorems of Chapter IV (which exclude the case oo — oo), (a) //(xO +h'{x.) = L /-iMzM^) + L /?M^MEl) (1) x=Zi X X] x^x\ X — Xi ^ ^ I h(x)-h{x^) ^ Hx)-U{x{) I ^ ^ /i(x)+/2(x)-/ifa)-/2(xO x-^x I X— a;i = L hix)- X- -/3(Xl) -a;i But by definition, /3'(Xl) = __^Hx)- -/3(a;i) -Xi (4) Henoe /3'(a;i) exists, and /a'Cxi) =/i'(xi) +/2'(xi). (&) h{x)=h(x)-Ux). Whenever Xt^xi we have the identity hix) -hjxi) _ /i(x) -Ux) -/i(xi) -/aCxi) X — Xi X — Xi _ /i(x) -/zCx) -/i(xi) -/aCx) +A(xi) -hjx) -/i(xi) -/zCxQ X — Xj =/2(a:) f/ i(x)-/i(xi) X— Xi 1 +,,(,) IteWa^l. "' t X-Xi J But the limit of the last expression exists as x=Xi (except perhaps in the case 00 — 00 ) and is equal to f2{x{)-U\x{)+h{x{)-U'{x{). 122 INFINITESIMAL ANALYSIS. Hence L ^;^ exists and fs'ixi) =h{xi) -h'ixi) +/2'(xi) -fiixi). (0 /^(^)=M^)- The argument is based on the identity /i(x)_/ifa) /2(X) /2(Xl) _ /l(x)-/2fa)-/2(x)-/lfa) , x-ii /2(a;) 72(2:1) -(x-xi) which holds when a; 7^X1 and when f2{x) 7^0. 3^^ fi(x)-f2(x^)-f2{x)-h(xr) f2{x)-f2{Xl)ix-Xi) Jljx) ■J2{x{)-h{x{) ■f2{x{)+h{Xi) ■U(Xl)-f2{x) -hJXi) U{x)-J2{Xi){x-Xi) /zfa) \h{x) -hjxi) I -/i(xi) {/2(x) -/aCzi) i f2{x)-f2iXl)ix-Xl) As before (excluding the case 00 — 00) we have , ,. , /2(Xl)-/l^fa)-/2^fa)'-/lfa) Corollary. — It follows from Theorems 74 and 75 of this chapter that if ]2{x)=a-fi{x) where /i'(x) exists, then /2'(x)=a-/i'(^). Theorem 76. Ifx>0, then -^x^ = A; • x*"i. DERIVATIVES AND DIFFERENTIALS. 123 (o) If A; is a positive integer, we have L — ^ — = L 1 a;*- 1 + x*^2. 3.J _|_ .,. 2.4.3.^4-2 +2.jjr-ij x=ii ■'' ^1 1=1, 171 {b) If A; is a positive rational fraction — , we have x"— Xi" \x"/ — XXi"/ L/ = Li JT ( J-'\"~l / J-^^n^Z ( 1\ ( l\n-l 2 i ~'\X^) + \X"/ • Vli"/ +. . .+ Ui"/ X"-Xi" 1 ( 1\'"-1 Tw:i-m\xi»/ , by the preceding case. 1 / J.\ "•""1 fn ——I But —T-^:^^-m\xi") =^^^1" =A;-Xi*-i. (c) If A; is a negative rational number and equal to — m, then, by the two preceding cases, ^3,^, X-Xi arix, a;"-!!™ X-Xi Xi^" = — Tnxi""*"^ But — TOXi~"'~^ = A;-x*"^ (d) If A; is a positive irrational number, we proceed as follows : 124 INFINITESIMAL ANALYSIS Consider values of x greater than or equal to unity. Let x approach Xi so that x>xi. Since, by Theorem 23, x'' is a monotonic increasing function of k for x>l, it follows that t^^^^^u.}^ y^.u'.SL X — Xi X — Xi X — Xi for all values of k' less than k, and all values of x greater than x^. If k' is a rational number, we have by the preceding cases that xY' KxJ -1 lii, X—Xi Since Xi*~^ is a continuous function of k, it follows that for every number N less than A;xi*~i there exists a rational num- ber ki' loss than k such that jV<A;i'-Xi*''-i<A;-2i*-i. Hence, by Theorem 40, xi«=- X — Xi cannot approach a value A^ less than A;xi*~i as x approaches Xi. By a precisely similar argument we show that a number greater than kxi''~'^ cannot be a value approached. Since there is always at least one value approached, we have that x'^-xi" If x<Xi as X approaches Xi, we write X*^ — Xi« -=x'' X — Xi X\—X and proceed as before. If A; is a negative number we proceed DERIVATIVES AND DIFFERENTIALS. 125 as under (c). The case in which a;i < 1 is treated similarly. For another proof see page 127. Theorem 77. ^ log„ x=-- log„ e. Proof. ^.2i?i^±^^)j:iog«_5=lw ^±i5 ix Jx ^ X =i..o.(l.f)r.. But, by Theorem 57, Therefore L — ^ — ^ ^2_ =, _ . i^g^ g_ Corollary. — log„ x = -. Theorem 78. // }i'(x) exists and if there is a V{xi) upon which fi(x) is continuous and possesses a single-valued inverse x = f2{y), then J2{y) is differentiable and If f'i^) is or +<x> or — co the convention + 00 — 00 is understood. Cf. Theorem 37. Proof. — To prove this theorem we observe that X = Xi ■^ ^l *=Ii ^ •''1 /l(x)-/i(Xi) By the definition of single-valued inverse (p. 45), x-X'^ _ hiy)-f2{yi) /i(x)-/(xi) y-yi t Theorem 78 gives a sufficient condition for the equality dx dx' dy 126 INFINITESIMAL ANALYSIS. Hence, by Theorems 38 and 34 and 37, x-x. x-xi u=m f2{y)-f2{yi) U'iy)' ]i.x)-f{xx) y-yi Theorem 79. // (1) /i'(x) exists and is finite for x^Xi, and fi(x) is continuous at x=Xi, (2) f2'iy) exists and is. finite for 2/1 =fi{xi), then ^/2{/i(xi)l=/2'(2/i)-/i'(2;i).t Proof. — We prove this theorem first for the case when there is a V*{xi) upon which fi(x) 7^/i(xi). In this case the following is an identity in x : /2{/l(x)i-/2l/lfa)i _ /2i/l(x)}-/2|/l(Xi)} /i(x)-/i(3:i) X-Xi fi(x)-fiixi) ' X-Xi By hypothesis (2) and Theorem 38, ' y^m y-yi xix. h{x)-fi{Xi) By hypothesis (1), Hence, by equation (1) and Theorem 34, we have the existence of ^fffC-rM r /2{/l(x)}-/2J/l(Xi)} f,..,,, . ^2|/i(a;)( = L^ ^3^^^ ■■f2{yi)-fi'(xi). If /i(x)=/i(xi) for values of x on every neighborhood of T=Xi, then, by hypothesis (1) and the corollary of Theorem 74, /'(xi)=0. t Theorem 79 gives a sufficient condition for the equality dz _ dz dy dx dydx' DERIVATIVES AND DIFFERENTIALS. 127 Let [x'] be the set of points upon which /i(a;)7^/i(xi). (There is such a set unless f(x) is constant in the neighborhood of x = xi.) Then, by the same argument as in the first case, we have d ^,/2l/i(a;i)! =/2'(j/i)-/i'(xi)=0 for X on the set [x']. Let [x"] be the set of values of x not included in [x']. Then a^Mh (xi) } - ,,L , —:^r—^ = 0' since the limitand function is zero. Hence both for the set [x'] and for the set [x"] the conclusion of our theorem is that the derivative required is zero. Theorem 80. d ^^a^- a- log a. Proof. — Let 2/ = a^, therefore log2/=x-logo dy and, by Theorem 77, dx . whence dy dx' = j/-log a = a* logo. This method also affords an elegant proof of Theorem 76, VIZ., d ;"=n2;"~'. Let J/=x", logj/=nlogx, dy dx _n y 'x' -r=n — =n-x"~i. dx X 128 INFINITESIMAL ANALYSIS. If § 3. Differential Notations. y = f{x) and L K, x=a ^ •*'! we denote f{x)-f{x{) hy Ay, and x-Xi by dx. Then, by Theorem 31, Mj^Ax-K+Ax- £{x), where Ax- s{x) is an infinitesimal with respect to Ay and Ax for x = a. This fact is expressed by the equation dy=K-dx, where K=j'{x). Here dy and dx are any numbers that satisfy this equation. There is no condition as to their being small, either expressed or implied, and dx and dy may be regarded as variable or Fig. 16. constant, large or small, as may be found convenient. When either dx or dy is once chosen, the other is, of course, determined. The numbers dx and dy are called the differentials of x and y respectively. DERIVATIVES AND DIFFERENTIALS 12& In Fig. 16, f{xi) is the tangent of the angle CAB, dx is the length of any segment AB with one extremity at A and parallel to the j-axis, and dy is the length of the segment BC. If x is regarded as approaching xi, then AB' is the infinitesimal ix, WD' is Jy, while Wc' is e{x) -Ax. Hence, by Theorem 73, 'WC' is an infimtesimal of higher order than Ax or Ay. We thus obtain a complete correspondence between deriva- tives and the ratios of differentials. Accordingly, for any for- mula in derivatives there is a corresponding formula in differ- entials. Thus corresponding to Theorem 75 we have : Theorem 8i. When for two junctions /i(x) and /zCx) dfi{x)=fi'{x)-dx and df2{x)=J2ix)-dx at Xi, it follows that (a) ///3(x)=/i(x)+/2(x), (km dfsixi) = |/i'(xi) +/2'(xi) \dx =d/i(xi)+d/2(xi). ib) ///3(x)=/i(x)-/2(x), then dU{x,) = \h'{x^)-U'ix{)\dx = d/i(xi)-d/2(xi). (c) // h{x)=h{x)-f2{x), then d/3(xi) = {/i(xi)-/2'(xi)+/2(xi)+/i'(xi)! -dx = /i(a:i)-d/2(xi)+/2(xi)-d/i(xi). , ^, , , \f2(x,)-h'{x,)-h(xi)-f2'ixr)\-dx then d/3(xi)= {f2{xi)\^ /2(x,)-d/i(x,)-/i(x,)d/2(xi) i/2(Xl)P The rule obtained on page 123 et seq. that the derivative of x* is Jfc • X*- 1 corresponds to the equation dx* = A; • x*- ^ • dx. If , in the 130 INFINITESIMAL ANALYSIS. equation dy=f{x)dx, dx is regarded as a constant while x varies, then dyisa. function of x. We then obtain a differential d2(dy) = {f'{x)-dx\d2X in precisely the same manner that we obtain dy=f{x)-dx. Since d2X may be chosen arbitrarily, we choose it equal to dx. Hence d{dy) =f"{x)dx^. We write this d'^y=f'(,x)-dx'^. The differential coefficient f"{x) is clearly identical with the de- rivative of fix). In this manner we obtain successively d^y=f^^\x)-dx^, etc. We may write these results, ^^fix) ^=f"(x) ^-n-Hx) dx '^^>' dx' ^ ^^^'•••-■dx"~' ^^''• Evidently the existence of the differential coefficient is coexten- sive with the existence of the derivative. § 4. Mean-value Theorems. Theorem 82. // /(x) has a unique and finite derivative at x = xi, then f(x) is continuous at x\. Proof.— The proof depends upon the evident fact that if f(x)-f{xi) approach anything but zero as x approaches Xi, then one of the values approached by m-f{x,) X — Xi is +00 or — 00 . Definition.— The function f(x) is said to have a maximum at x=xi if there exists a neighborhood V(xi) such that (1) No value of fix) in 7(xi) is greater than /(xi). (2) There is a value of x, X2, in 7(xi) such that X2<xi a.nd/(x2)</(ii). DERIVATIVES AND DIFFERENTIALS. 131 (3) There is a value of x, 13, in V{xi) such that X3>a;i and /(X3)</(Xi). Similarly we define a minimum of a function. This definition allows any point of a constant stretch like a, Fig. 17, to be a maximum, but does not allow any point of b to be either a maximum or a minimum. Fig. 17. Theorem 83. // fixi) exists and if /(i) has a maximum or a minimum at x=xi, then f{xi) =0. Proof. — In case of a maximum at xi, it follows directly from the hypothesis that rix. X-Xi < xix, X-Xl X>Xi *<^1 Since f(xi) exists these limits are equal, that is, the derivative is equal to zero. Similarly in case of a minimum. Theorem 84. // /(xi)=/(x2), fix) being corUimwus on th£ 132 INFINITESIMAL ANALYSIS. interval Xi X2, and if the derivative exists t at every point between Xi and X2, then there is a value f between Xi and X2 such that /'(?)= 0. The derivative need not exist at Xi and X2- Proof. — (a) The function may be a constant between Xi and X2, in which case f{x) = for all values of x between Xi and X2 by Theorem 74. (6) There may be values of the function between Xi and X2 which are greater than f{xi) and f{x2). Since the function is continuous on the interval Xi X2, it reaches a least upper boimd on this interval at some point X3 (different from Xi and '0:2). By Theorem 83, /'(X3)=0. (c) In case there are values of the function on the interval I — I Xi X2 less than /(xi), the derivative is zero at the minimum point in precisely the same manner as under case (b). Fig. 18. This theorem is called Rolle's Theorem. The restriction that /(x) shall be continuous is unnecessary if the derivative t Not necessarily finite. DERIVATIVES AND DIFFERENTIALS. 133 exists, but simplifies the argument. The proof without this restriction is suggested as an exercise for the reader. The geometric interpretation is that any curve representing a continuous fimction, }{x), such that f{xi) =f{x2), and having a tangent at every point betweeen xi and X2 has a horizontal tangent at some point between them. An immediate gener- alization of this is that between any two points A and B on a curve which satisfies the hypothesis of this theorem there is a tangent to the curve which is parallel to the Une AB. The following theorem is a corresponding analytical generali- zation : I — I Theorem 85. // /(x) is continuous on the interval. Xi X2, and if the derivative exists at every point between X\ and X2, then there is a value of x, x = $, between xi and X2 such that ' X1-X2 Proof. — Consider a function /i(z) such that fii.x)=f{x)-{x-x2) — ^^_^^ ; then fi{xi)=fix2) and /i(x2)=/(x2). Therefore /i(a;i)=/i(x2). Hence, by Theorem 84, there is an x, x = $ on the segment xi X2 such that /i' (0=0. That is, />'©=/'(« -'-^^^-0. Therefore f(f)-^^^^^. This is the "mean-value theorem." Its content may also be ■expressed by the equation f{X2)=f{Xl) + {X2-Xl)n^). 134 INFINITESIMAL ANALYSIS. Denoting xi -x by dx and f by x + Odx, where 0<5<1, it takes the form /(xi +dx) =/(xi) +f (xi + ddx)dx. Theorem 86. // /i(x) and /2(x) are continiious on an interval a b, and if /i'(x) and /a'Cx) exist between a and b, fz'ix) 5^ ± oo , and /2'(x)7^0, /2(a) 5^/2 (&), ^^len there is a value of x, x=f between a and b such that fi(a)-fi( b)_fi'{?) /2(a) -/2(6) /2'(f)- Proof. — Consider a function Since /3(a) =0 and fsib) =0, we have as before /a'Cf) = 0. Therefore AMzMl^M) inereiore /2(a) -/2(6) /a'Ce)" This is called the second mean-value theorem. The first mean-value theorem has a very important extension to "Taylor's series with a remainder," which follows as Theorem 87. § 5. Taylor's Series. The derivative of f{x) is denoted by /"(x) and is called the second derviative of /(x). In general the nth derivative is the derivative of the n - 1st derivative and is denoted by /^"'(x) . Theorem 87. // the first n derivatives of the function f{x) l-l exist and are finite upon the interval a b, there is a value of x, x„ l-l on the interval a b such that DERIVATIVES AND DIFFERENTIALS. 135 /(&)=/(a)+^/'(a)+^V(a) + . ^ (n-1)! ' ^'^^+ „, ; {Xn). Proof. — Let i?„ be a constant such that Fix) =f{x) -fia) - {x-a)na) J-^^f"{a) -... _ (x-a)"-^ ,„_.) (.T-g)" (n-1)! ' '■'^^ n! "" is equal to zero for x=b. Since F{x)=0 for x = a, there is, by Theorem 84, some value of x, Xi,a<xi<b such that i^'(xi)=0. That is, F'ix)=nx)-na)-{x-a)r{a)-. . . _ (x-a)"-' , (x-a)"-> (n-2)! ^ "^"^ („_i), «n is equal to zero for x=a;i. Since also F'{a) =0, there is a value of X, X2, a<X2<Xi such that F"{x2)=0. Proceeding in this manner we obtain a value of x, x„, a<x„<Xn-i such that F^^KXn)=0. But F">(x„)=p(a;n)-ii;n=0. Therefore /i!„ = f"H2:»), whence the theorem. Corollary— In Theorem 87, f^'^Kx) need be supposed to exist only on ab. Definition. — The expression n! n- *"=o "'• is called the remainder, and the infinite series t-o «• is called Taylor's Series. 136 INFINITESIMAL ANALYSIS. a constant different from zero, » /W.(g)(b-g)n then 2 —^ n=0 "• is convergent but not equal to j(b), i.e., If L^-(&-a)» fails to exist and be finite, then 00 /('')(a) „=o «■! (6 -a)" is a divergent series. Hence an obvious necessary and sufficient condition that for a function /(x) all of whose derivatives exist for the values of I, o<x<6, n=0 "• is that L '— V^ (6 -a)» = O.f This leads at once, by Theorem 33, to the following sufficient condition : n—00 ni for every value of i on o b is not sufficient, since in depends upon n. DERIVATIVES AND DIFFERENTIALS. 137 Theorem 88. // /(")(x) exists and |/(")(x)| is less than a fixed I — I quantity M for every x on the interval a b and for every n {n = l,2, . . .), then m=m + ^^f'ia) + . . . + ^i^;(n)(a) + . . . Functions are well known all of whose derivatives exist at I — I every point on an interval a b, but such that for some point on this interval n = "'' where fi is a function of x not identically zero. Other func- tions are known for which the series is divergent. The classical example of the former is that given by Cauchy,t e~? at the point x = 0. If this function is defined to be zero for 2 = 0, all its derivatives are zero for x=0, whence Taylor's development gives a function which is zero for all values of x. Pringsheim X has given a set of necessary and sufficient conditions that a function shall be representable for the values oi h, 0<h<R,hy means of the series l-,./W(0)-A". It was remarked above, p. 131, that a necessary condition for f(x) to be a maximum at a; = a is f (a) =0 if the derivative exists. Taylor's series permits us to extend this as follows : Theorem 89. // on some V{a) the first n derivatives of f{x) exist and are finite and on V*{a) /'"■•■ '^(x) exists and is bounded,^ and if ■|- Cauchy, Collected Works, 2d series, Vol. 4, p. 250. j A. Pringsheim, Mathematische Annalen, Vol. 44 (1893), p. 52, 53. See also Koxig, Mathematische Annalen, Vol. 23, p. 450. § Instead of assuming the existence of /<»+')(x) we might have assumed A") (a:) continuous without essentially changing the proof. 138 INFINITESIMAL ANALYSIS. 0=/'(a)=/"(a) = ..;=/(»-i>(a), then : (1) If n is odd, f{x) has neither a maximum nor a mini- mum at a; (2) // n is even, j{x) has a maximum or a minimum according as f'^aXQ or f''\a)>Q. Proof. — By Taylor's theorem, for every x in the vicinity of a fix) =/(a) + (x-o)«/(")(o) + (a;-a)"+i-/(»+»(f«), where ^x is between x and a. Hence j{x) -f{a) = (x- a) "{/(») (a) + (i-a)/("+i)(f ^) } . But since /^""""^'(fi) is bounded and x—a is infinitesimal, there exists a 'V*{a) such that if x is in 7* (a), Kx)-m is positive or negative according as (x-a)"-/(")(o) is positive or negative. (1) If n is odd,(a;— a)"is of the same sign as x—a, and hence for /("'(a) >0 j{x)-f{a)>0 '\ix>a, f{x)-J{a)<0 iix<a; while for f"Ka)<0 /(x)-/(a)>0 ifa;<a, /(x)-/(a)<0 ifx>a. (2) If n is even, (x—a)" is always positive, and hence if f"'(a)>0. /(x)-/(o)>0 ifa;>a, /(x)-/(a)>0 ifx<a; then /(a) is a maximum. DERIVATIVES AND DIFFERENTIALS. 139 If /(">(a)<0. /(x)-/(a)<0 if a;>a, 1 tr \ t/ \ ^n -t ^ f then /(a) is a mimmum. /(x)-/(a)<0 if i<a; J '^ -^ § 6. Indeterminate Forms. The mean-value theorems have an important application in the derivation of l'Hospital's rule for calculating "indeter- minate forms." There are seven cases. fix) (1) -, i.e., to compute L -tt-t if L /(x)=Oand L <}>{x)=0. (2) — , i.e., to compute L -rr^ if L f{x) = ± oo and L 0(1) = ±00. (3) 00 —CO, i.e., to compute L j/(x)— ^(i)| if L f(x)= ±00 x=a x^a and L <f>{x)= ±00. (4) O-oo, i.e., to compute L /(x)-0(i) if L /(x)=0 and Z/ <f>ix)= ±00. (5) 1°°, i.e., to compute L /(x)*(^> if L /(x) = l and L 4>(x) = ± 00 . (6) 0°, i.e., to compute L /(x)*^^) if L /(x) = and L ^(x) =0. x=a x=a x^a (7) 00 0, i.e., to compute L f{xY^''^ if L/(x)=±oo and L 0(x)=O. These problems may all be reduced to one or the other of the first two. The third may be written (since /(x) t-^O on some V*{a)) /(i)-<^(x)=-^ ^(x) = — ^. W) Jixj which is either determinate or of type (1). 140 INFINITESIMAL ANALYSIS. To the cases (5), (6), and (7) we may apply the corollaries of Theorem 39 of Chapter IV, from which it follows (provided fix) 7^0 on some F*(o)), that exists if and only if log L /(x)*(^>= L log/(a;)*(^>= L 9&(a;) log/(x) exists. x=a x=a x^a The evaluation of L -~ ^) comes under case (1) or case (2). The evaluation of cases (1) and (2) is effected by the follow- ing theorems: Theorem 90. // /(x) and 4>{x) are continuous and differentiable and <^(x) is monotonic and (f>'{x) ^0 and <j>'(x) j^ 00 and (1) i] L f{x)=0 and L <j>ix)=0 or (2) if L ,^(x)=±oo,t XwOO 0(x) exists and is equal to K. Proof .—For every positive h we have, by the second mean- value theorem, fix +h)- fix) _f(e.) <t>ix+h)-<}>ix) <f>'i$^y where ^^ lies between x and x + h. But since f ,, takes on values which are a subset of the values of x, and since L f = 00 t It is not necessary that Lf{x)=aa ■ cf. Theorem 69. DERIVATIVES AND DIFFERENTIALS. 141 which in turn implies L 77 r{ — -~rT = K x^oo<p{x + h)-<j>{x) ' and this, according to Theorems 68 and 69, gives Corollary. — If f{x) is continuous and differentiable, then L — L fix). The theorem above can be extended by the substitution 1 z = - x-a to the case where x approaches a finite value o. The approach must of course be one-sided. Theorem 91. // fix) and (f>{x) are continuous and differen- tiable on same V*(a) and f{x) is hounded on every finite interval, while (f>{x) is monotonic and (1) L /(x)=0, L 0(1) =0 or x=a x—a (2) L ^(x) =+ 00 or - 00 : thenxf xt¥(i) ' it follows that L 77-r exists and is equal to K. 142 INFINITESIMAL ANALYSIS. fix) Proof. — If L ,,, . exists, the limit exists when the approach x=a V \-^) is only on values of x>a. Consider only such values of x. Then if ^=^' /(^)=/(«+7)=-P'(2) and 4>{x)='^{a+-)=0{z), by hypothesis and Theorem 79, F'iz) and 0'iz) exist and i^'(3)=/'(x)g. Hence If LVi^r^' \ hen, according to Theorem 38, r F'iz) exists and is equal to K. Hence, by Theorem 90, exists and is equal to K. T ZM .t^<I>iz) fix) Hence, by Theorem 38, L ., . exists and is equal to K. We have now derived conditions under which we can state a general rule for computing an indeterminate form. Provided fix) is not zero on every V*ia), any of the forms <3) to (7) can be reduced to Fix) (Pix) W DERIVATIVES AND DIFFERENTIALS. 143 where this is of type (1) or (2). Provided Fix) and 0{x) satisfy the conditions of Theorem 91, the existence of the limit of (a) depends on the existence of the Umit of F'ix) ¥{xj- (^) If (6) is indeterminate, and F'{x) and 0'{x) satisfy the condi- tions of Theorem 91, the limit of (6) depends on the Umit of F"{x) '(X) 'ft-y\> 'W and so on in general. If at each step the conditions of Theorem 91 are satisfied and the form is still indeterminate, the Umit of i?'(n+l)(x) depends on the limit of ^u+iv^ ('i+l) If (n) is indeterminate for all values of n, this rule leads to no result. If for some value of n then all the preceding Umits exist and are equal to K, and so x=o<P(a;) The original expression is equal to X or e^ according to the case under consideration. 144 INFINITESIMAL ANALYSIS. § 7. General Theorems on Derivatives. Theorem 92. // /(x) is coniiniious and f(x) exists for every x on an interval a b, then f'{x) takes on every value between any two of its values. Proof. — Consider any two values of f'(x), f'{x\), and f'ixi) on the interval a b. Consider, further, the function X — Xi on the interval between Xi and X2. Since is a con- X — Xi tinuous function of x on this interval, it takes on every value between and /'(xi), which is its limiting value as x approaches xi. Hence, by Theorem 85, /'(x) takes on all values between and including f{xi), and — — for values of x I — I on the interval Xi X2. By considering in a similar manner the /(X2) -fix) i — I function on the interval Xi X2, we show that f'{x) takes on all values between - — ^-— ^ — — and /'(X2). Hence /'(x) takes on all values between f(xi) and /'(X2). Theorem 93. // the derivative exists at every point on an interval, including its end-points, it does not follow that the de- rivative is continuous or that it takes on its upper and lower bounds. Proof. — ^This is shown by the following example. The curve shall he between the x-axis and the parabola 2" —1 j/ = ix2. The straight lines of slopes 1, IJ, If, . . . , 1 h — - — . . . 2" through the points (i, 0), (i, 0), . . . , [^ifi, o) , . . ., respectively, meet the parabola in points Ai, A2, A3, ... , A„, . . . The broken line ^1 (J, 0) A2 (i, 0) A3 . . .An {^,0j . . . oo , has an DERIVATIVES AXD DIFFERENTIALS. 145 infinitude of vertices. In each angle of the broken line con- sider an arc of circle tangent to and terminated by the sides of Fig. 19. the angle, the points of tangency being one fourth of the distance to the nearest vertex. The function whose graph consists of these circular arcs and the portions of the broken line between IH them is continuous and differentiable on the interval 1. Its derivative is discontinuous at j=0 and has the least upper bound 2, which is never reached. Theorem 94. // /'(x) exists and is equal to zero for every valiie l-l , . , of X on the interval a h, then f{x) is a constant on thai interval. Proof. — By Theorem 82, fix) is continuous. Suppose /(i) not a constant, so that for two values of x, ii, and X2, f{xi)7if{x2), then, by Theorem 85, there is a value of x, x=$ between xi and X2 such that ^^^^' X2-X, ' 146 INFINITESIMAL ANALYSIS. which is different from zero, whence f'(x) is not zero for every 1-1 value of X on the interval a b. Hence f(x) is a constant on l-l a b. Corollary. — If /i'(x) ^fzix) and is finite for every value of x on an interval a b, then fi{x) —J2(x) is a constant on this interval. Theorem 95. // j'ix) exists and is positive for every value l-l of x on the interval a b, then f(x) is monotonic increasing on this interval. If f'{x) is negative for every value of x on this interval, then f{x) is monotonic decreasing. Proof, — If f{x) is positive for every value of x, then it fol- lows from Theorem 85, provided that f{x) is continuous, that the function is monotonic increasing, for if there were two values of x, Xi and X2, such that /(xi) t /(X2) while Xi <X2, then there would be a value of x, x = f , between xi and X2 such that /'(f) _ /(^2)~/fa) <o X2-X1 In case fix) is not supposed continuous, the argument can be made as follows: If f{xi)>0, then, by Theorem 23, there exists about the point xi a segment (xi— iJ), (xi-f-<J), upon which /(£W(£l)>0, X — Xi and hence, if x>Xi, /(x)>/(xi) and if x<X], /(x)</(xi). Now about every point of the segment a b there is such a segment. Let xf and x" be any two points of a 6 such that x'<x". By Theorem 10, there is a finite set of these segments of lengths Si . . . dn which include within them every point of the interval I — I x' x". We thus have a finite set of points, namely, the mid- point and points on the overlapping parts of the segments x'<Xi<X2<... <Xft<x", such that DERIVATIVES AND DIFFERENTIALS. 147 /(a/X/CxiX/feX. . .</(xfc)</(x"). Hence /(x')</(x"). In a similar manner we prove that the function is monotonic decreasing in case f{x) is negative. Theorem 96. // a function f{x) is monotonic increasing on l-l an interval a b, and if f{x) exists for every value of x on this interval, then there is no point on the interval for which f{x) is negative. That is, f{x) is either positive or zero for every point I — I of a b. Proof. — If /'(x) is negative for some value of x, say Xi, then L = C, a negative number, x=xi X — Xi whence there is a neighborhood of Xi on which /(x) >/(xi), while x<xi, or /(xi)>/(x), while x>Xi, \Yliicli is contrary to the hypothesis that the function is monotonic increasing in the neighborhood of x = ii. In the same manner we prove that if the function is monotonic decreasing, and if the derivative exists, then fix) cannot be positive. The following theorem states necessary and sufficient condi- tions for the existence of the progressive and regressive deriva- tives. Conditions for the existence of a derivative proper are obtained by adding the condition that the progressive and regressive derivatives are equal. Theorem 97. // /(x), x<Xi, is continuous in some neighbor- hood of x=Xi, then a necessary and sufficient condition that f{xi) shall exist and be finite is that there exists not more than one linear function of x, ax+c, such that /(x) +ax+c vanishes on every neighborhood 0/ x = xi. Proof__(l) The condition is necessary. We prove that if /'(x) exists and is finite, then not more than one function of the form ax + c exists such that /(x) +ax+c vanishes on every neighbor- hood of x=Xi. If no such function exists, the theorem is veri- fied. If there is one such function, the following argument wUl show that there is only one. Since, by hypothesis, 148 INFINITESIMAL ANALYSIS. exists, we have, by Theorem 75, that f{x)+ax+c-j{xi)-axi-c exists. Let [a/] be the subset of the set of values of x on any neighborhood of x=xi such that j{x')+ax!+c vanishes on the set [a/]. By Theorem 41, j{xf)+axf +c-f(xi) -axi -c ^ ^f(x)+ax+c-f(xr)-ax,-c ^^^^^^_^^^ X=Xi ■^ -^1 Since /'(xi) and a are both finite, f{x')+ax^+(/-f{xi)-axi-c x'^xi x! -Xi is finite. But the numerator of this fraction is a constant, f{x)+ax+c being zero on the set [x']. Hence ^ /(x)+ax+c-/(x0 .3ax^c^ or /'(xO+a=0, xix, X-Xi and, being continuous, /(xi)+axi+c=0. The numbers o and c are miiquely determined by the equations i/'(xi)+a=0, (/(xi)+axi+c=0. (2) The condition is sufficient. We are to show that DERIVATIVES AND DIFFERENTIALS. 149 j- /W-/(Xi) can fail to exist only when there are at least two functions of the form ax+c such that /(x)+ax + c vanishes on every neigh- borhood of i=xi. If L Mz/(£l) x-xi X — Xi does not exist, then ^S^^-J.^^ X — Xi approaches at least two distinct values Ki and K2. Let K2<Ki. Let A and B be two finite values such that K2<A<B<Ki. On every neighborhood of x = xi there are values of x for which X-Xi is greater than B, and also values of x for which f(x)-f{xi) X-Xi is less than A. Hence, since /(x)-/(xi) X-Xi is continuous at every point except possibly xi, in a certain neighborhood of Xi there are values of a; in every neighborhood of xi for which =A, X — Xi or fix)-}{xi)=A(x-Xi), which gives -f(xi) -A{x-xi) as one function of the form ax+c. 150 INFINITESIMAL ANALYSIS. In the same manner we show that —f(xi)—B(x—Xi) is another function ax+c, which makes f{x)+ax+c vanish on every neighborhood of x=^xi. The geometric meaning of this theorem is obvious. If P is a point on the curve representing f(x), then a necessary and sufficient condition that this curve shall have a tangent at P is that there exists not more than one line through P which inter- sects the curve an infinite number of times on any neighborhood of P- Compare the fvmctions x sin — and x^ sin — on page 51. The earlier mathematicians supposed that every continuous function must have a derivative except at particular points. The first example of a fimctipn which has no derivative at any point is due to Weiers iwadc . 7^ The function is 00 /(x)= I 6" cos (a'^nx), where a is an odd integer, < 6 < 1 and ai>>l + |?r. t For references and remarks see page 51 . CHAPTER VIII. DEFINITE INTEGRALS. § I. Definition of the Definite Integral. The area of a rectangle the lengths of whose sides are exact multiples of the length of the side of a unit square, is the num- ber of squares equal to the unit square contained within the rectangle, and is easily seen to be equal to the product of the lengths of its base and altitude.t In case the sides of the rectangle and the side of the unit square are commensurable, the sides of the rectangle not being exact multiples of the side of the square, the rectangle and the square are divided into a set ..of equal squares. The area of the rectangle is then defined as the ratio of the niunber of squares in the rectangle to be measxared to the number of squares in the unit square. Again, the area is equal to the product of the base and altitude. Any figure so related to the unit square that both figures can be divided into a finite set of equal squares is said to be com- mensurable with the miit. The area of a rectangle incommensurable with the unit is defined as the least upper boimd of the areas of all commensur- able rectangles contained within it. It foUows directly from the definition of the product of irrational nvunbers that this process gives the area as the prod- uct of the base and altitude. J t Of course the units are not necessarily squares; they may be triangles, parallelograms, etc. X For the meaning of the length of a segment mcommensurable with the unit segment, compare Chapter II, page 33. 151 152 INFINITESIMAL ANALYSIS. Turning to the figure bounded by the segment a b (which we take on the x axis in a system of rectangular coordinates) the graph of a function y^fix) and the ordinates a;=a and x=6, Fig. 20. we obtain as follows an approximation to the common notion of the area of such figures. Let xo=a, xi, X2, . . . , Xn = b he a. set of points lying in order from a to 6. Such a set of points is called a partition of a h, and is denoted by n. The intervals xq xi, xi Xz, . . . , a;„_ii„ are intervals of ;:. Let xi— zo=.^ia;, X2—xi=J2X, . . . , x„-Xn-i=^„x, and let fi, $2, . . ■ , $n be a set of points such that $1 is on the interval xq Xi, $2 is on I 1 . I 1 Xi X2 . • • , and $n IS on i„_ix„. Then /(fi), fiU), ..., /(f„) are the altitudes of a set of rectangles whose combined area is a more or less close approximation of the area of our figure. Denote this approximate area by S. Then S=m)Jix+m)J2X + . . .+/(f„)i„x= I fih)^kX. k-l As the greatest Ji^ is taken smaller and smaller, the figure DEFINITE INTEGRALS. 153 composed of the rectangles comes nearer to the figure bounded by the curve. In consequence of these geometrical notions we define the area of the figure as the limit of S as the J^x's decrease in- definitely. The area S is the definite integral of /(x) from a to 6. It has been tacitly assumed that the graph of y=f{x) is continuous, since we do not usually speak of an area being enclosed by a discontinuous curve. The definition of the defi- nite integral when stated in its general form admits, however, of functions which are discontinuous in a great variety of ways. A more general definition of the definite integral is as follows : 1 — I I — I Let a h {or h a) he an interval upon which a function f{x) is defined, single-valued and bounded. Let n, stand for any par- I — I I — I tilicn of a b or b a by the points a = Xo, Xi, X2, . . . , x„ = 6 such that the numbers A\X = Xi—a, A2X = X2 — X\,..., J„x~6— x„_i are each numerically less than or equal to d. fl, f2, ■ • • , fn be a set of points on the intervals I 1 I — I I 1 , ., ^ I 1 I 1 I 1 Xq-Xi, Xi X2, ■ . . , Xn-lXn (pr if b<a, Xi Xo, X2 Xi, X3 X2, I 1 . . . , x„ x„_i) respectively, and let // the many-valued function of d, S„ approaches a single limiting value as d approaches zero, then L S,= f){x)dx. When we desire to indicate the interval' of integration we write \Si and *?:, instead of Ss and n,. a and b are called the limits of integration. The details of this definition should be carefully noted. 154 INFINITESIMAL ANALYSIS. For every d there is an infinite number of different partitions TTg, and for every partition there is an infinite set of different sets of f A, so that for every d the function Sg has an infinite set of values. The graph of the function S» is of the type shown in Fig. 21. Every value of S» for one d is assumed by *S for every larger d. For any particular value of d the values Fig. 21. of Si lie on a definite interval BS3 BS,, whose length never in- creases as d decreases. If this interval approaches as 5 ap- proaches 0, the required hmit exists. It is to be noticed that the set of ;r's, [:r*] includes every possible n whose largest JkX is less than d. Thus we carmot obtain the set of all ;r's by sequential repartitioning of any given n, since there are partitions of the set [ng] which have no partition points in common with any given partition. Inat- tention to this point is perhaps the greatest source of error in the development of the notion of a definite integral. § 2. Integrability of Functions. The class of integrable functions is very large, including nearly all the bounded functions studied in mathematics and DEFINITE INTEGRALS. 155 physics. Even such an arbitrary function as y=Qii X irrational, y=^\/n^ if x=m/n, is integrable. (See page 182, Theorem 127.) Examples of non-integrable functions are y=\/x on the interval 1 (where it is not bounded, see page 191), and the function, 2/ =0 if a; is irrational and y=\\i x\s rational. To determine the conditions of integrability we introduce the concept of integral oscillation. On any interval a b, f(x) has a least upper bound A and a greatest lower bound B, be- tween which the function varies. If A—B=Jy=''Ofix) is mul- tiplied by the length of the interval, Jx = \b-a\, it gives the area of a rectangle, including the graph of f{x). If the interval is subdivided by a partition ;:, the sum of the products Jx-Jy on the intervals of the partition is called the integral oscilla- tion of fix) for the partition tt and is denoted by 0^. If we call A]cy the difference between the upper and lower bounds of /(x) I 1 on the intervals Xk-iXk, we have n Q^ = \Aix\-Aiy + \A2x\-A2y-\-... + \Anx\Jny-=I\Akx\-dky. k= 1 Greometrically 0,, represents the areas of the rectangles Fi,...,F^ (Fig. 22), and so we exp)ect to find that if the lower bound of 0, is zero, fix) is integrable. This proposition, which requires some rather deUcate argument for its proof, will be taken up in § 7. At present we shall show in a simple manner that every continuous and every mono tonic function is integrable. Lemma i. If S„ and SJ are two sums (formed by using differ- ent $k's) on the same partition, then 156 INFINITESIMAL ANALYSIS. Proof. l'S,--S,'| = I{f{^k)'f{^k')\JkX 4=1 ^ ^ i/(fft)-/(.V)|-Mjfcx|. A=l But |/(f jt) -/(f*')l^ifc2/ by the definition of Jty. Therefore |5„-S/|< I \Jkx\-dky (4) k=l ( f^^\ A,!/ V ; T h a \ 1 Fry/ -\-^ Fio. 22. A repartition of a partition ;r is formed by introducing new points in t:. Lemma 2. // tti is a repartition of n, Proof. — Any interval Jita; of ;: is composed of one or more DEFINITE INTEGRALS. 157 intervals Jk'x, dk"x, etc., of n^, and these contribute io S„ the terms K^k')dk'x+f{^n^k"x+ (1) The corresponding term of *S, is /(ftM*x=/(fO^/a;+/(fi)J/'x + (2> But since |/(fft)-/(f«:') 1=^*3/, the difference between (1) and (2) is less than or equal to Aky-\dk'x^Ak"x + . . .\^Aky-\dkx\ and hence \S:,-SA< I Aky-U,^\=0^. ' k=\ I Theorem 98. Every Junction continuous on a b is integrable I — I on a b. Proof. — ^We have to investigate the existence of the limit LS} of the many-valued function Ss as d~0. Since S, ap- proaches at least one value as d approaches zero (see Theorem 24), we need only to prove that it cannot have more than one value approached. Suppose there were two such values, B R — C and C, B>C. Let £ = — —. By the definition of value approached, for every 8 there must exist an S (which we call Sb) such that \SB-B\<e, (1) and such that the corresponding ttb has its largest AkX<8. Similarly there must be an Sc such that \Sc-C\<^, (2) and such that the corresponding nc has its largest JkPO<d. Let TT be a partition made up of the points both of tcb and nc, and let S be one of the corresponding sums. ;r is a repartition both of TZB and nc. 158 INFINITESIMAL ANALYSIS. Therefore |5-<Scl<0;rp (3) and |S-5B|<ar^ (4) But since f(x) is continuous, by the theorem of uniform conti- nuity, d can be so chosen that if any two values of x differ by less than d, the corresponding values of /(x) differ by less than and hence on the partitions ttb and nc, whose J^x's are all £ \b-a\ less than 8, the corresponding i^j/'s are all less than ,, _ , . So n we have (since I AkX = h—a) On = i" \JkX\-Jky< - |itx|-|T-— T = £• Hence 0„„<£ and On„<s. B—C So we have, since e = — j— and d is so chosen that whenever \^-x"\<d,\f{x')-fixf')\<~^^ \Sb-B\<s, \Sc-C\<e, \Sb-S\<b, \Sc-S\<e. From these inequalities it follows that \B—C\<4e, which con- tradicts the statement that «= — j—. Hence the hypothesis that fix) is not integrable is untenable. Theorem 99. Every non-oscillating hounded function is integrable. Proof. — The proof runs, as in the preceding theorem, to the DEFINITE INTEGRALS. 159 paragraph following (4). Let D and d be the upper and lower bounds of /(z) . d, being arbitrary, can be so chosen that 8 = jr-^. D—a Then 0„^= 3 Jky-\^kx\< I /t^y-S, ° k-l A=l and since f{x) is non-oscillating, idky=D-d. Therefore 0„ „ < (D - d) 5 = e. Similarly Ojt < e. Hence again we have \SB-B\<e, \Sc-C\<^, \Sb-S\<^, \Sc-S\<^, and therefore |5— C|<4e, whereas e was assumed equal to R — C — -7 — . Thus the hypothesis of a non-integrable non-oscillating fimction is imtenable. § 3. Computation of Definite Integrals. In computing definite integrals it is important to observe that whfn the integral is known to exist the Umit can be cal- culated on any properly chosen subset of the Si's. (See Theorem 41.) So we have that if <S,„ S)^, ... is any sequence of sums such that L 5„=0, then L Sin ttsX One case of this kind occurs when f * is taken as an end- 160 INFINITESIMAL ANALYSIS. point of the interval Xk-i Xk and all the i^x's are equal. Then we have rb n I / f{x)dx= L I f{a+kJx)Jx, -where Ax = - 6— a A simple example of this principle is the proof of the following theorem . Theorem loo. // ]{x) is a constant, C, then rCdx=CQ)-a). Proof. — The function f{x) = C is integrable either according to Theorem 98 or Theorem 99. Hence rCdx= L I C^—= L n-C^-^^=CQ)-a). A few other examples follow. In each case the function is known to be integrable by the theorems of § 2. /}'' Theorem loi. / e^dx=e''— e". Proof. — =e^-Jx-—7- — 7-'- — 7^ — r^'^-^x . , t. •. ^^ Ax Whence the result follows since L -j- — :=1. (Differentiate JiioC — i mmierator and denominator with respect to Jx according to Theorem 90. DEFINITE INTEGRALS. 161 Instead of arranging the partition-points in an arithmetical progression as in the cases above, we may put them in a geo- metrical progression, that is, we let 6\" 6 aj ==«' a =3"' Aix = aq — a, J2X = aq^—aq, . . . , J„x = ag"— 05""^, fi=a, 62=03, . . . , f„=a5"-i, and obtain the formula rfix)dx= L a{q-lMa)+qfiaq)+... +q'^-'f(.aq--^)] J <• « = i = L a{q-l)'l qi^fiaq^). jii 4=0 We apply this scheme to the following. Theorem 102. In all cases where m is a whole number?^ -1, and if a>0, b>0 for every value of m?^ -1, / 6 ^m+1 (j'n+1 x'^dx = ■ m + 1 X^dx= L a{q-l) I q''{aq'')'" a 9=1 i-0 =a'"+i L (3-l)[l + (r+i) + (r+^)' + - ■ . + ir^')^-n (1) (om+l)n_l = La'"+M(3")"'^'-i!jFrri 9 = 1 ^ 162 INFINITESIMAL ANALYSIS. Hence / x^dx = - m + 1 ' T g-1 1 Theorem 103. / -dx = log b—loga, {0<a<b). Proof. By equation (1) in the last theorem, since g™+i = g° = 1 , / -dx= L niq-1); >°^ a) but n =— i , hence log 3 £V^= ,il^- ^°S (^)=log (|)=log&-loga, since (§6, Chapter VII) l'Hospital's rule gives 8=1 log g The following theorem is of frequent use in computing both derivatives and integrals. I — I Theorem 104. // on an interval a b two functions /(x) and Fix) have the ■property that for every two valries of x, xi and X2, ■where a<xi<X2<b, fiXl){X2-Xi) < F{X2) -F(Xi) < /(X2)(X2-Xi); or if /(Xi)(X2-Xi) > i^(X2) -2^(Xi) > /(X2)(X2-Xi), then (1), if f{x) is coniiniums, DEFINITE INTEGRALS. 163 and (2) whether f{x) is continuous or not, I }{x)dx exists and is equal to F{h) —F{a). Proof. — We consider first the case fiXi)iX2-Xi)<F{X2)-F{Xi)<fiX2){X2-Xi). This gives /(xi) < — - — < /(X2). X2 — Xi Since f{x) is continuous at x = xi, L /(x2)=/(xi). Hence, by Theorem 40 (Corollary 2), F(x2)-Fixx) ^ — 77^:7, — =/(a;i), zj-l, X2— Xl which proves (1). To prove (2) we observe that f{x) is non-oscillating and therefore integrable according to Theorem 99. On any parti- tion 7t whose dividing points are xi, X2, . . . , x„_i we have /(o)(xi-a) <F{xi)-Fia) </(xi)(xi-a), /(Xi)(X2-Xi) <F{X2)-FiXi) </(X2)(X2-Xi), /(Xn_i)(5-x„_i) <F(b) -F(x„_i)</(6)(6-x„_i). Adding, we get /(o)(xi-a)+/(xi)(x2-xi) + . . .+/(x„_i)(6-x„_i)<F(6)-J!?'(o) ^/(a;i)(2;i-a)+/(x2)(x2-xi) + . . .+f(jb)ib-Xn-l). But /(o)(xi-a) +. . .+/(x„_i)(6-x„_i)>BS, and /(xi)(xi-a)+. . .+/(b)(&-x„_i)<B<S,. 164 INFINITESIMAL ANALYSIS. Since this holds for every n, we have by Theorem 40 that as (Theorem 99) / f{x)dx exists, /^<' x)dx=Fib)-F{a). The proof in case fixi){x2-Xi)>F{x2) -Fixi)>f(x2){x2-xi} is identical with the above when we write > instead of <. § 4. Elementary Properties of Definite Integrals. Theorem 105. // a<b<c, and if a bounded function f{x) is integrable from a to c, then it is integrable from a to b and from b to c. Proof. — Suppose f(x) not integrable from a to b, then .by the definition of a limit (see Chap. II.) there must be a set of values of IS3, [aS/], such that L 18,'= A, and another set [aS/'] such that L 183" = B, while A and B are distinct. Whether }=0 £ f{x)dx exists or not, there must be a set of values of ISt^ [iSi'l such that the limit L IS/ =C. Now for every »S/ and iS/ there exists a IS/ such that lS/=iS/+tS/. Therefore A+C is a value approached by IS}. By similar reasoning, 5 + C is a value approached by IS}- Hence IS 3 has two values approached, which is contrary to the hypothesis. Hence / {x)dx must exist. By similar reasoning / f{x)dx must exist. Theorem 106. // a<b<c and if a bounded function f{x) is integrable from a to b and from b to c, then f{x) is integrable from f{x)dx= / f{x)dx+ I f{x)dx. Proof. — Since / f{x)dx and / f{x)dx exist, we know by Theorem 26 that for every e there exists a ^c' such that for DEFINITE INTEGRALS. every value of *5, where d < d/, aSi- I f{x)di <S' and also a d." such that for every value of tSi where d<d/', ts^-fj( 'x)dx £ l-l 165 (1) (2) Now if the upper bound of /(x) on a c is M and its lower bound is m, let oj" ■■ -, and let d, be smaller than the smallest of 5/, dg" , d,'". Consider any value of IS,. If the point b is one of the points of the partition upon which ISa is computed, then tS, is the s.im of one value of iS, and one value of IS,. If b is not a point of this partition, let JbX be the length of the inter- val of a-^i that contains b. Then for properly chosen ^Ss and bS, \%+lS,-'aSs\<AbX{M-m)<-^. ... (3) So in evory case (whether or not 6 is a partition-point of „;:,) by combining (1), (2), and (3) we obtain the result that for every e there exists a d, such that for every aSi, dx\<e. IS,,- fj{x)dx- fj{x)dji Therefore L 'aS,= / /(x)dx+ Imdx, which proves the theorem. Theorem 107. Provided both integrals exist, ■f and a<b, r\m\dxt\ f){x)dx . t That the first integral exists if Che second exists is shown in Theorem 135. 166 INFINITESIMAL ANALYSIS. Proof. I\}i^k) MiX > I Ifi$k)^kX\. Hence for every (Sj|/(x)| there is a smaller or equal Si fix), the d's being the same. Hence by CoroUary 2, Theorem 40, LS,\m\>\ LSem\. Theorem io8. // / f{x)dx exists, then I f{x)dx exists and f f(x)dx=- ffixjdx. Proof. — ^This is a consequence of the theorem (Corollary 1 Theorem 27) that L (-/(x)) = - Lfix), a sum cor- for to every S used in defining / f{x)dx corresponds equal to —S which is used in defining / f{x)dx. Similarly to every S' used in defining / f{x)dx there responds a sum -S' used in defining / fix)dx. Hence the function Sg in the definition of / f{x)dx is the negative of the function Sg used in the definition of / f{x)dx. Hence the theorem follows from the theorem quoted. We adjoin the following two theorems, the first of which is an inmiediate consequence of the definition of an integral, and the second a corollary of Theorems 105, 106, and 108. DEFINITE INTEGRALS. 167 Theorem 109. J ^^^ f{x-h)dx exists and is equal to / f{x)dx, provided the latter integral exists.\ Theorem no. // any two of the following integrals exist, so does the third, and £ f{x)dx+ Jjf{x)dx= r f{x)dx. Theorem in. If C is any constant and if f{x) is integrable I — I I — I onab, then Cf{x) is integrable on ah and £'cf{x)dx = cj'''fix)dx. n Proof.— Sa= I fi$k)^kX is an S) of the set which defines i— 1 fb J^ f{x)dx and 5/= I^ Cf{$k)J^ is the corresponding S» of the set which defines / Cf{x)dx. Hence our theorem follows immediately from Theorem 34, a special case of which is L Cf{x) =CLf{x). Theorem 112. // fi{x) and fzix) are any two functions each I — I integrable on the interval a b, then f{x) =f\{x) ±f2{x) is integra- I — I bU on ah and rf(x)dx= rfi{x)dx± Pfzixjdx. Proof. — ^The proof depends directly upon the theorem that if L ^i(x)=6i, and L ^(x) =62; then L ^i(x)±^(x)=6i±62 x=a x=a x=a (Theorem 34). t First stated formally by H. Lebesode, Lemons smt VInUgration, Chapter VU, page 98. 168 INFINITESIMAL ANALYSIS. 1-1 Theorem 113. // /i(x) and foix) are integrable onab and such l-l that for every value of x on a b fi(x)tf2{x), then rf,(,x)dx> rf2{x)dx. Proof. — Since Si is always greater than or equal to S2, then, by Theorem 34, L Sit L S2, which proves the theorem. 1=0 oiO Theorem 114. (Maximum-Minimum Theorem.) If (1) the product fi{x)-f2{x) and the factor fi{x) are inte- i-l grable on a b, I 1 (2) /i (x) is always positive or always negative on a b, (.3)' M and m are the least upper and the greatest lower I — I bounds respectively of /2(x) on a b, then m- / fi{x)dx< / fi{x)f2ix)dx<M ■ I fi{x)dx, or m- f fi{x)dx > f)i{x) -fz^x >M- P fx{x)dx. Proof. — By Theorem 111, M ■ J fi{x)dx== rM-fi{x)dx and m- / fi{x)dx= I m-fi{x)dx. %/ a */o But in case /i(x) is always positive, m-fi{x)<fi{x)-f2{x)<M-fi{x). Hence, by the preceding theorem. J^m-fi{x)dx ^fjiix) ■f2{x)dx <f^M.f{x)dx, DEFINITE INTEGRALS. 169 and therefore m- rji(x)dx< rh{x)-f2ix)dx<M- P U{x)dx. J a %J a J a If/i(x) is always negative, it follows in the same manner that m- rji{x)dxl /^fi{x)-h{x)dx>M- Ai(i)dx. As an obvious corollary of this theorem we have the Mean- value Theorem : Theorem 115. Under the hypothesis of Theorem II4 there exists a number K, m'S K^M, such that rh{x)-f2(x)dx=K f''h{x)dx. Corollary 1. In case f^^x) is continuous we have a value I — I f of X on a h such that Ai(:c)-/2(x)dx=/2(f) f U^)dx. J a 'J "■ In case /i(x)=l, / /i(x)dx = 6-a, and the theorem reduces to this : Theorem 116. // /(x) is any integrable function on the inter- val a b, there exists a number M lying between the upper and lower I — I bounds of /(x) on a b such that rfix)dx = M{b-a), I — I and if fix) is continuous, there is a value $ of x on a b swk that fj{x)dx = fmh-a). 170 INFINITESIMAL ANALYSIS. In many applications of the integral calculus the expression Cmdx —^ represents the notion of an average value of the dependent variable y=f{x) as x varies from a to 6. An average of an infinite set of values of /(x) is of course to be described only by means of a limiting process. Consider a set of points \-. — I xj, X2, . . . , Xn-i,Xn=b ou thc interval a b such that Xi—a = X2—Xi = X3—X2 = . ■ . = X„-i—X„^2=b—Xn-l. Then M„=- I f{xk), and we define the mean value of f{x), lM}(x)= L M„ if this n=oo ,. . . T% b — a linut exists. But Xk+i—Xk= =ix. If the definite integral / f{x)dx exists, we may write i/ a / f(x)dx= LS», where Si= I f{xk)Ax= I f{xk)^^ = ^^^ 1 f{xk) = (b-a)M„. k-i k"! n n jc^i Therefore L Sa = ib-a) L M„. } = n=ixi We therefore have the theorem: Theorem 117. In case the integral of f{x) exists on the interval yf{x)dx a o b, iMfix) = b — a We note that \M is the same as the K which occvirs in the mean-value theorem, and that the last theorem suggests a simple DEFIXITE IXTEGRALS. 171 method of approximating 'the value of a definite integral by multiplying the average of a finite nmnber of ordinates by b-a. § 5. The Definite Integral as a Function of the Limits of Integration. I 1 Theorem 118. // /(.r) i\>f integrahle on an intenal a b, and I — I r^ if X i^ any poini of a b, I f{x)dx is a contimtous function of x. Proof. — / /(,x)dr exists, b>- Theorem 105, and by the defini- tion of a continuous function we need only to show that L^(^fj{x)dx-fjKx)dj^ =0. By the theorems of the preceding section, A(j)dr - A(x) rf-r = rf{x)dx<\iB- (/ -x) \<\B- [x' -x)\, %/a U a *J I where {B stands for the least upper bound of f{x) on the inter- I — 1 - I — I val J x! , and B for the least upper bound of f{,x) on a b. Smce 5 is a constant, B\x' -x) approaches zero as x' approaches x. and therefore by Theorem 40, Corollary 4, the conclusion of our theorem follows. Theorem 119. // /(x) Vs conXinuoM^ on an inten'ol a b, /fix)dx (a<x<6) possesses a derivative icith respect to x such that Proof.— By the preceding theorem / /(x)dx is continuous. 172 INFINITESIMAL ANALYSIS. To form the derivative we investigate the expression f)(x)dx- f){x)dx f){x)dx X' —X x' —X as x' approaches x. By Theorem 115 (the mean-value theorem), (1) f%)dx = mx!)){xf-x), where f (x) is a value of x between x and x' and is a function of x'. Hence (1) is equal to m (2) But as xf approaches x, f also approaches x and so, by Theorem 39, as x' approaches x, (2) approaches j{x). Therefore / l{x)dx- / j{x)dx , -=/(x)4/«x)i.. X —X Following is a more general statement of Theorem 119. Corollary. — ^If /(x) is continuous at a point Xi of a b ana mtegrable on a b, then at a; = Xx lf)(x)dx^Kx). The proof is like that of Theorem 112 except that y/(x) dx = {x- Xi)M(x) , and M(xi) is a value between the upper and lower bounds of DEFINITE INTEGRALS. 173 fix) on xi X. But by the continuity of f{x) at i L Mix) = fix,), and hence the conclusion follows as in the theorem. Theorem 120. // fix) is any continuous function on the inter- I — I val a b, and Fix) any function on this interval such that im-fix), then Fix) differs from I fix)dx at most by an additive constant. Proof.— Let Fix) = j ^fix)dx + 4>ix). Since Fix) and / fix)dx are both differentiable, By the preceding theorem d, dx. fjix)dx=fix). Hence j-^ix) =0, whence, by Theorem 94, <^(a;) is a constant. Theorem 121. // fix) is a continuous function on an interval a b and Fix) is such that then £fix)dx = Fib)-Fia). /; 174 INFINITESIMAL ANALYSIS. Proof. — By the last theorem, f{x)dx=F{x)+c. But 0= rf{x)dx=F{a)+c. Therefore -F(a)=c. Whence f''f{x)dx=F{b) +c=i?'(6) -F(a). The symbol [F{x)fa or |S F{x) is frequently used for ^(6) -F{a). In these terms the above theorem is expressed by the equation ){x)dx = \lF{x). r By this last theorem the theory of definite and indefinite integrals is united as far as continuous functions are concerned, and a table of derivatives gives a table of integrals. For dis- continuous functions the correspondence does not in general hold. That is, there are on the one hand integrable functions ](x) such that / \{x)dx is not differentiable with respect to x, .and on the other hand differentiable functions ^(x) such that ■^'(x) is not integrable. t § 6. Integration by Parts and by Substitution. The formulas for integration by parts and by substitution are ordinarily written as follows: / udv = uv— / vd.u, f M^y- J Kyy%dx. t For a good discussion of this subject the reader is referred to H. Le- BESGUE, L&;ons sur V Integration. DEFINITE INTEGRALS. 175 The following theorems state sufficient conditions for their validity. Theorem 122. (Integration by parts.) £ hix) ■h'{x)dx = \jM ■f2{x)^- fj2{x) ■h'{x)dx, -provided //(x) and fz'ix) exist and are continuous on the interval I — I a b. Proof. — By Theorem 75, £{hix) -Mx)) =/i(x) -k'ix) +k(.x) ■h'i.x). Therefore X di^f^^""^ •/2W)dx = y^ /i(x) ■f2'{x)dx + £''j2{x) ■fi'ix)dx. (The integral exists since it follows from the existence and continuity of //(x) and ^'(a;) that /i(x) and /zCz) are continuous). By Theorem 121, £ii\h(x) •Hx)\dx=hQ>) ■f2ib) -hia) ./2(a). Therefore fjl(x) ■h'{x)dx = [hix) ■hix)l- fj2{x) ■ll'(x)dx. Theorem 123. (Integration by substitution.) If y = cp(x) has a I— I continuous derivative at every point of ab and f(y) is continuous for all values taken byy = (j)(x) as x varies from a to b, where A = (j>(a), B = <f>(b). 176 INFINITESIMAL ANALYSIS. Proof.— By Theorem 120 and by Theorem 79, C being an arbitrary constant. C is determined by letting X = a. Then if x = 6 we have* Theorem fjmy-£mt-d-- f(x)dx = J^ fi4>iy))-^dy, where x — <j){y) and a = 4>{A), h = 4>{B); provided that both inte- grals exist, and that <f>{y) is non-oscillating and has a finite derivative. Proof. f f{x)dx= L I f{$k)^kX (1) whenever the least upper bound of J4X for each n approaches „ , B-A zero as n approaches + 00 . Now let jy= , yk=A+k-Jy, 4>(yk)-4>(.yk-i)-.dkX. Hence, by Theorem 85, AkX = ^'(j)]^Ay, where ij* lies between j/i and yk-i- Now if fj: = ^(ijA), it will lie between <j){yk) and cj>{yk-i); moreover the Akx's are all of the same sign or zero; and since the hypothesis makes ^(t/) uniformly continuous, their least upper bound approaches zero as n approaches + 00 . Therefore f f{x)dx= L I f{^k)^k£ = L I f{4>irik))-<f>'{r,k)-Jy nioo k=l ^fii>iyW(.y)dy, DEFINITE INTEGRALS. 177 provided the latter integral exists. Hence fji'')^^ = fj('f>iy)) '^dy. Corollary. — The validity of this theorem remains if <f>(y) has a finite number of oscillations. Proof. — Suppose the maximum and minimum values of <p{y) are Oi, (h, 0,3, ... , o„, corresponding to the values of y, Ai, A2, A3, ... , An. Then we have ffix)dx= f''){x)dx+ f\x)dx + ...+ rf{x)dx r^ dx =//(*(x);5* The form of this proposition given in Theorem 123 would per- mit an infinitude of oscillations of ^(2/). § 7. General Conditions for Integrability. The following lemmas are to be associated with those on pages 155 and 156. Lemma 3. If ttj is a repartition of n, then for any function I — I bounded on a b 0.,<0.. Proof. — Any interval iti of ^ is composed of one or more intervals Ak'x, Ak'x, etc., of ni, and these contribute to 0,, the terms \Ak'x\Ak'y + \Ak"x\Ak"y + (1) 178 INFINITESIMAL ANALYSIS. The corresponding term of 0^ is \Jkx\Jky = \^k'x\J^y + \Jk"x\Jky+ (2) Since each of JVy, ^k"y, etc., is less than or equal to Jky, (1) is less than or equal to (2), and hence 0,. ^ 0». I — 1 Lemma 4. If kq is any partition of the interval a b, and eo any positive number, then for any bounded function there exists a number do such that for every partition tc whose greatest J is less than 80 Proof. — We prove the lemma by showing that if no has N + 1 partition points Xo, Xi, X2, . . . , x„, an effective choice of ^0 is where R is the oscillation of the function on a b. Of the intervals of tz there are at most JV — 1 which con- tain as interior points, points oi xq, xi, . . . , xif. Denote the lengths of these intervals of n by JpX, and denote by JgX the lengths of the intervals of k which contain as interior points no points of Xo, Xi, X2, . . . , x^. Then V q If ;:' is a repartition of no obtained by introducing the points of n, then Q is a subset of the terms whose sum constitutes O^/. Hence, by Lenuna 3, i'|i^|.i,2/<0.-<0,„. Since \A^\<s^ = J2-^ DEFIXITE INTEGRALS. 179 it follows that i'|Jpx| • Jp2/<«o. p Therefore 0,„ + £o^O,. Lemma 5. If ;r is any partition, 0, is the least upper bound of the expression o» — »S, , where SJ and SJ' may be any two values of 5, corresponding to different choices of the f 's. Proof. — ^Without loss of generality we may assume every Jtx positive. Then BS^ -BS, = B\S: -S^"\. But BS, = b\ I K^k)-A,A =• I {5(/et)|Jtx It-i J *=i and BS^ = B- t=i J *=i - Therefore BS^-BS,= -" [5/(ft)-J3/(ft)]Jtx — *-i — n Therefore 5(5/ -5/') =0.. Theorem 125. A necessary and sufficient condition that a function f{x), defined, single-valued, and bounded on an interval I— I I— I a b shaU be integrable on a b, is that the greatest lower bound of On for this function shall be zero. Proof. — ^We first show that if /(x) is integrable the lower bound of 0, is zero. By hypothesis, r^f{x)dx= L S) ■exists. By Theorem 27, Chapter IV, this implies that for every s 180 INFINITESIMAL ANALYSIS. there exists a S, such that for every di<d. and 32<d, Hence, if ;r be a partition whose intervals i^x are all less than d„ we inust have \SJ-S/'\<e ''•! for every 5/ and S/'. By Lemma 3 this implies that 0„ < s. But if for every £ there exists a tt such that 0^ < e, then 50. =0. Secondly, we show that if the lower bound of 0. is zero, S3 converges to a single value, f){x)dx, as d approaches zero. Given any positive quantity s there exists a partition n, such that On,<-7. By Lemma 4 there exists a S, such that for every ti whose intervals are numerically less than d, 0,<0..+|<-|. Now let Sn,' and Sn," be any two values of Sg^, and let ;:/" be the partition composed of the points of both ;r/ and 7:,". Then for any value of Sn^"' we have, by Lemma 2, \S„; -Sn."'\ 20< <^, \Sn,"-S„;"\<On;'<-^. Therefore \S,t,' -Sn,"\<s. DEFINITE INTEGRALS. 181 Hence for every e we have a d, such that for every two values oiS,,d<d„ \S.,'-Sn,"\<e. By Theorem 27, this imphes the existence of L Sg. In case the definite integral does not exist it is sometimes desirable to use the upper and lower bounds of indeterminate- ness oi Ssas d approaches zero. These are denoted respectively by the symbols / f{x)dx and / }{x)dx f and are called the upper and lower definite integrals of /(x) They are both equal to f)ix)dx if and only if the latter integral exists. They are usually defined by the equations BS. f f{x)dx=R -'it; «/ a where 5,= i" {Bfi^^)\Jia for all partitions of tc, and / fix)dx = BS, where S,= 1 i^/Cc^) UhX for all partitions of n. k=l That f f{x)dx exists when the upper and lower integrals are equal is evident under this definition, because o.=s^-s^, t For a more extended theory of these integrals, cf. Pierpont, page 337. 182 INFINITESIMAL ANALYSIS. and thus B0^=0 if and only if /* fix)dx= f fix)dx. For every value of 5>0 there is an infinite set of partitions n, for which the largest J^a; is less than d, and for each of these there is a value of 0^. If Os stands for any such 0^, then 0) is a many-valued function of b. Theorem 126. A necessary and sufficient condition that a function f{x), defined, single-valued, and hounded on an interval I — I a b, is integrable is that L O,=0. Proof.T-r^e condition is necessary. By Theorem 125 the integrability of f(x) implies B0„=0. Hence for every e there exists a partition t: such that By Lemma 4 there exists a 1?, such that for every s^ whose greatest Jx is less than d, 0,'<0, + £<2s. Hence L O' = 0. The condition is sufficient. Since L O*=0, andO,>0, B0],=0. Hence the function is integrable by Theorem 125. Theorem 127. A necessary and sufficient condition that a function, defined, single-valued, and bounded on an interval a b, shall be integrable on that interval is that for every pair of positive DEFINITE INTEGRALS. 183 numbers a and X there exists a partition n such that the sum of the lengths of those intervals on which the oscillation of the function is greater than a is less than A. Proof. — The condition is necessary. If for a given pair of positive numbers a and X there exists DO ;r such as is required by the theorem, then 0^> a- A for every n, which is contrary to the conclusion of Theorem 125. that The condition is sufficient. For a given positive e choose a and X so that e £ a(b — a)<-^ and XR<-^, where R is the oscillation of the function on a b. Let s^ be a partition such that the sum of the lengths of those intervals on which the oscillation of the function is greater than a is less than A. Then the sum of the terms of 0^ which occur on these intervals is less than XR, and the sum of the terms of 0^ on the remaining intervals is less than aQ) — a) . Therefore On<X-R + aQ)-a)<£. Hence BO, = 0, whence by Theorem 125 the integral exists. Defimtion.— The content of a set of points [x] on an interval a 6 is a number C[x] defined as follows: Let w be any parti- tion of a b, none of the partition points of which are points of [x], and D, the sum 'of the lengths of those intervals of n 184 INFINITESIMAL ANALYSIS. which contain points of [x] as interior points. Then BD^ = C[x]. An important special case is where C[x] = 0. It is evident that if a set [x] has content zero, for every e there exists a finite set of segments of lengths «i, «2, £3, • • • , «n which contain every point [x] and such that n 1=1 It is also evident that if the sets [xi] and [2:2] are of content zero, then the set of all Xi and X2 is of content zero.f Theorem 128. A necessary and sufficient condition for the integrability of a function f(x) on an interval a b is that for every a>0 the set of points [x„] at which the oscillation of f{x) is greater than or equal to a shall he of content zero.X Proof. — If at every point of an interval c d the oscillation of /(i) is less than a, then about each point of c d there is a segment upon which the oscillation is less than a, and hence by Theorem 11, Chapter II, there is a partition of c d upon each interval of which the oscillation of f{x) is less than a. Now to prove the condition sufficient we observe that if the content of {x„] is zero, there exists for every X a partition TZi such that the sum of the lengths of the intervals containing points of [x,] is less than X. Moreover we have just seen t For further discussion of the notion content see Pierpont, Real Func- tions, Vol. I, p. 352, and Lebesque, Lemons sur I'lntigration. X Compare the example on page 155. DEFINITE INTEGRALS. 185 that the intervals which do not contain points on [x„] can be repartitioned into intervals on which the oscillation is less than a. Hence, by Theorem 127, the function is integrable. To prove the condition necessary we note that on every interval containing a point, x„ the oscillation of j{x) is greater than or equal to or equal to a. Hence, if C[xj>0, the sum of the intervals upon which the oscillation is greater than or equal to ct is greater than C[x<,]. Definition. — A set of points is said to be numerable if it is capable of being set into one-to-one correspondence with the positive integral numbers. If a set [x] is numerable, it can always be indicated by the notation Xi, X2, X3, . . . , Xn, . . . , or \xn\, but if it is not numerable, the notation \xn\ cannot be apphed with the understanding that n is integral. Theorem 129. A 'perfect set of points is not numerably infinite.^ Proof.— Suppose the theorem not true. Then there exists a sequence of points \xn\ containing every point of a perfect set [x]. Let Pi be any point of [x], and a^ bi a segment containing Pi. Let x„, be the first of lx„! within oi 61. Since x„ is a limit point of points of [x], there are points of the set other than Pi and z„, on the segment ai bi. Let P2 be such a point, and let 02 62 be a segment within ai 61 and containing P2 but neither Pi nor x„,. Let x„^ be the first point of {x„i within 02 62. Proceeding in this maimer we obtain a sequence of segments ja, bi\ such that every segment lies within the preceding and such that every segment ai bi contains no point ^ni-ic of the sequence \xn\. By the lemma on page 42, Chapter II, there is a point P on every segment of this set. Since there are points of [x] on every segment a,- bi, P is a limit point of the set [x]. Since [x] is a perfect set, P is a point of [x]. But if P t For definition of perfect set see page 91. 186 INFINITESIMAL ANALYSIS. were in the sequence jz„}, there would be only a finite number of points of [x] preceding P, whereas by the construction there is an infinitude of such points. Theorem 130. A numerably infinite set of sets of points each of content zero cannot contain every point of any interval. Proof. — Let the set of sets be ordered into a sequence {[x]„] . We show that on every segment a b there is at least one point not of ![x]„). Since [x]i is of content zero, there is a segment oi 61 contained in a 6 which contains no point of [x]i. Let [x]„j be the first set of the sequence which contains a point of Ci bi. Since [x]„, is of content zero, there is a segment 02 62 contained in ai 61 which contains no point of [x]„,. Continuing in this manner we obtain a sequence of segments o b,ai 61, ... , a„ bn . . . such that every segment lies within the preceding, and such that a„ 6„ contains no point of [x]i, . . . , [x]„. By the lemma on page 42 there is at least one point P on all these segments. Hence P is a point of a 6 and is not a point of any set of |[x]„!. Theorem 131. The points of discontinuity of an integrabk function form at most a set consisting of a numerable set of sets, each of content zero. Proof. — Let <ti, 02, 0-3, .. . be any set of numbers such that and L a„ = 0. n=oo By Theorem 128 the set of points [x„J at which the oscillation of f{x) is greater than or equal to On+i and less than «;„ is of content zero. Since the set of sets j[x„Ji includes all the points of discontinuity of fix) , this proves the theorem. Theorem 132. // a function f{x) is integrable on an interval I— I o b, then it is continuous at a set of points which is everywhere I— I dense on a b. DEFINITE INTEGRALS. 187 Proof. — ^If the theorem fails to hold, then there exists an interval a 6 on which the function is discontinuous at every point. By Theorem 131 an integrable function is discontinuous at most on a numerably infinite set of sets each of content zero, and by Theorem 130 such sets of sets fail to contain every point of any interval. Theorem 133. // / f{x)dx = for every X of a b, then /(x)=0 on a set of points everywhere I — I dense on a b, and for every a>0 the points where \fix)\> a form a set of content zero. Proof. — ^At every point X where f{x) is continuous, accord- ing to the corollary of Theorem 119, smce I f{x)dx IS a constant. The points of continuity of f{x) J a are everywhere dense, according to Theorem 132. Hence the zero points of /(i) are everywhere dense. At a point of discon- tinuity the oscillation of f{x) is greater than or equal to |/(x)|. Hence the points where |/(x) | > a form a set of content zero. Theorem 134. // rx nx I f{x)dx= I (f>{x)dx I 1 for every X of a b, then f{x)=<f)(x) on a set of points everywhere I — I dense on a b, and for every <7>0 the points where \f{x) —<f){x)\>a forms a set of content zero. Proof. — Apply the theorem above to /(x) -<^(x). Theorem 135. // /(x) is integrable from a to b, then |/(x)| is integrable from a tob.'f tThe converse theorem is not true; cf. example given on page 192. 188 INFINITESIMAL ANALYSIS. Proof.— Since 0<0,\Kx)\<OJ{x), it follows that BO^f{x) = implies 5 0„|/(x)| = 0, and hence the integrabilityof j{x) implies the integrability of |/(a;)|. Theorem 136. // /(x) and <l>{x) are both integrabk on an inter- I— I vol a b, then f{x)-4>{x) (1) I— I ^ - IS integrable on a b; and, provided there is a constant m > such I — I that |^(x)| — m>0 for x on a b, then /(x)H-^(x) (2) I— ! is integrable on a b. Proof. — Since f{x) and ^(x) are both integrable on a 6, it follows that for every pair of positive numbers a and A there is a partition tti for fix) and a partition 712 for ^(x) such that the sums of the lengths of the intervals on which the oscilla- tions of f{x) and <f>(x) respectively are greater than a are less than X. Let 7: be the partition consisting of the points of both TTi and 712. Then the sum of the intervals of tz on which the oscillation of either f{x) or (p{x) is greater than a is less than 2A. Let M be the greater of B\f{x)\ and B\(f>{x)\ on a b. Then on any interval of 71 on which the oscillations of /(x) and (p{x) are both less than a the oscillation of /(x)-^(x) is less than aM. Hence the sum of the intervals on which the oscillation of /(x) ■ 0(x) is greater than aM is less than 2A. Since a and A may be chosen so that 2 A and aM shall be any pair of preassigned numbers, it follows by Theorem 127 that 1—1 fix) ■<f>ix) is integrable on a b. In view of the argument above it is sufficient for the second DEFINITE INTEGRALS. 189 part of the theorem to prove that -t-~. is integrable on a 6 if </>(a;) is integrable and \(f>{x)\ >m. Consider a partition re such that the sum of the intervals on which the oscillation of ^(x) is greater than a is less than X. Since 1 1 ^(Xi) <j>{X2) .. \<f>(Xi)~(<j>X2)\ mXl)\-\<f>{X2)\ it follows that n is such that the sum of the intervals on which the oscillation of -ri-^ is greater than —5 is less than X, and T7-T is integrable according to Theorem 127. A second proof may be made by comparing the integral oscillations of /(x) and ^(x) with those of the functions (1) and (2) and applying Theorem 125. j Theorem 137. // /(x) is an integrable function on an interval 1—1 a b, and if <p{y) is a continuous function on an interval Bf Bf, where Bf and Bf are the lower and upper bounds respecl- ~ ~~ I — I ively of f(x) on a b, then (p\f{x)\ is an integrable function of x I — I on the interval a b.X Proof. — By Theorem 48 there exists for every a>0 ad„ such that for \yi-y2\<d„ \<f>(yi)-<f>(y2)\<<T (1) Since f(x) is integrable on a 6 it follows by Theorem 127 that for every positive number X there is a partition ic such t Cf. PlERPONT, Vol. 1, pp. 346, 347, 348. X This theorem is due to Dtr Bois Retmond. It cannot be modified so as to read " an integrable function of an integrable function is integrable." Cf. E. H. Moore, Annals of Mathematics, new series, Vol. 2, 1901, p. 153. 190 • INFINITESIMAL ANALYSIS. that the sum of the intervals on which the oscillation of f{x) is greater than d„ is less than X. But by (1) this means that the sum of the intervals on which the oscillation of ^{f{x)\ is greater than a is less than A. This, by Theorem 127, proves that ^l/(x)l is integrable. CHAPTER IX. IMPROPER DEFINITE INTEGRALS. § I. The Improper Definite Integral on a Finite Interval. If fix) is infinite at one or more points of the interval a b, then, whatever may be the other properties of the function, the definite integral of /(x) defined in Chapter VIII cannot exist on I — I the interval a b. Definition. — ^If / f{x)dx exists for every x,a<x<b, and if f L rfix)dx x=aJ T. exists and is finite, /(i) being unboimded on every neighbor- hood of x = a, then this limit is the improper definite integral I— I on the interval a b. If f{x) is unbounded m every neigh- borhood of x=a, and also in every neighborhood of x=b, but bounded on some neighborhood of every other point of the I— I . . I— I I— I interval a b, we consider two intervals a c and c b where c is any point a<c<b. If the improper definite integral exists I— I 1—1 , . . on a c and also on c b, then the sum of these mtegraJs is the I— I improper definite integral on a b. t We will understand throughout this chapter that in the expression (,x)dx I 1 X approaches a on the interval o 6. 191 192 INFINITESIMAL ANALYSIS. This definition can obviously be extended to the case where the function is unbounded in the neighborhood of a finite num- ber of points. Such points are then considered as partition points, dividing the interval a b into a set of subintervals. If the improper definite integral exists on each of these in- I— I tervals, their sum is the improper definite integral on a b. Theorem 138. // / f{x)dx exists for every x, a<x<b, then a necessary and sufficient condition that L rf{x)dx t=aJx shall exist and be finite is that for every e there exists a V,*{a) I — I such that for every two values of x, Xi and xz, on the interval a b and on V,*{a) I rx, \J f^' |m/ Xi x)dx <s. Proof. — This theorem is a special case of Theorem 27, since, by Theorem 110, f{x)dx= / f{x)dx- / f(x)dx. Xi *J X\ \J X2 Theorem 139. // / f{x)dx exists for every x, a<x<6, and if L r\f{x)\dx x=a«/ X is finite, then L rf(x)dx exists and is finite.^ t The first part of the hypothesis in this theorem is not redundant as is shown by the following example. Let f{x) = x-i for positive rational values IMPROPER DEFINITE INTEGRALS. 193 Proof.— By the necessary condition of Theorem 138 there is a V*{a) corresponding to any preassigned e such that for any two values of x, xi and Xz, which lie on the segment a~b and on V*{a) I r)fix)\d2 <£. But, by Theorem 107, \ r)f{x)\dx\>\ r){x)dx\, since, by the hypothesis and Theorem 105, f^^f{x)dx exists. Hence, by the sufficient condition of Theorem 138, L f)ix)d2 exists and is finite. Theorem 140 I. // / ]{x) dx exists for every x on the segment a b, and if (x—a)''f(x) is bounded on V*(a) for some valve of k, 0<k<l, then L rf{x)dx x=aJ X exists and is finite. Proof. — By hypothesis (i— a)*|/(a;)|<Af, i.e., of I and /(i)=— 2~J for positive irrational values of x. In this case L I \f(x)\dx exists and is finite, while / j{x)dx does not exist for any value of X on the interval o 5, and consequently L I f(x)dx has no mean- xiavx ing since the limitand does not exist. 194 INFINITESIMAL ANALYSIS. where M may be taken greater than one. The proof of the theorem consists in showing that for every e there exists a d, such that if 0<xi-a<d„ 0<X2-a<d„ Xi<X2, then I r^"' I f{x)dx < e. By Theorems 105 and 133, =^^\ix2-ay-'-(xi-ay-''\. That the last term of this series of inequalities is infinitesimal, the reader may verify by choosing This theorem may also be proved as a corollary of Theorem Corollary. — If /(x) is integrable on x 6 for every x of a 6, 1 is of the same oi k, 0<k<l, then 143 and is of the same or lower order than -. rr for some value {x — a)" L Prndz exists and is finite. Theorem 141. // for any positive number m and for any ktl there exists a V*{a) on which f{x) does not change sign, and on ■which (x— o)* /(x) >m for every x, then L rf{x)d2 ^aJ X cannot exist and he finite. L IMPROPER DEFINITE INTEGRALS. 195 Proof. — (1) In case /jix)dx fails to exist for some value of x between a and b, L f''fix)dx fails to exist because the limitand function does not exist. (2) If fj^''^'^- exists for every value of x between a and h, we proceed as follows : Let 5< 1 be the length of a V*{a) on which f(x) does not change sign, and on which {x — aYj{x)>m, and let Xi be the extrem- ity of this neighborhood, which is greater than a. Then l/(^)l>(^r^>(^^r^* ^^^ ^^^'■y ^ ""^ *^^ neighborhood. Take xi so that (X2 - a) * = 2(a;2 - Xi) . >fe?^^^^-^^) = ^- Then / f{x)dx Hence, by the necessary condition of Theorem 138, L rf{x)dx cannot exist and be finite. Theorem 142. // L f{x)dx cxiste and is finite and if fix) approaches infinity monotonicaliy as x=a on some V*ia), then L (x-a)-/(x)=0, 196 INFINITESIMAL ANALYSIS. 1 or in other words f{x) has an infinity of order lower than ^3^-t Proof.— By means of Theorem 138 it follows from the hypoth- esis that for every £ there exists a 7* (a) within V*{a) such that for every Xi and X2 on a b, and also on V*ia), / fix)dx <£. Let X2 be any point of such a neighborhood and let Xi be so chosen that Xi — a = X2 — Xi . Since Xi and X2 are on V*{a), /(Xi)>/(X2). It follows from Theorem 116 that But / fix)dx >|/(X2)|-(X2-Xi). /(X2) • (X2-X1) = J/(X2) • (x2-a). f L (i— o)-/(x) =0 is not a sufficient condition for the existence of L I l(x)dx, xia'' X as is shown by the following example. Consider a set of points i,, I2 •Cs, . . . , i„, . . . such that Xn — a = 2(i„-|-i — a), Xi — a being unity. 2" Define /(i,) = l, /(x2)=|, /(i,)=2, . . . , /(=r")=^:fri' Letthefunc- tion be linear from f{xO to /(12), from /(xj) to /(xa), etc. Then 1/ Wx\ /(x)dx >i, /(x)<ix >J, etc. Since these integrals are all of the same sign, their sum for any given num- ber of terms is greater than the sum of the corresponding number of terms. 2 in the harmonic series. Also (x„ — o) ■ /(xr.) = — tT' ■'^l»e°ce L (x — a)/(x)=0. IMPROPER DEFINITE INTEGRALS. 197 Hence for x = X2, \j{x) \-{x-a)<2e. Since e is arbitrary, and since X2 is any point in V*{a), it follows that L /(x)-(x-a)=0. Corollary. — If / f(x)dx exists for every x between a and h, and l^ f)ix)dz L x=a\J X exists and is finite, and if /(x) is entirely positive or entirely negative, then zero is a value approached by {x — a)-f(x) as x approaches a. Proof. — Consider the case when the function is entirely positive. Suppose zero is not a value approached. Then there exists a pair of positive numbers £ and 8 such that for every X, x — a<d, {x—a)-f{x)> £. I— I On the interval, a a + d, consider the function x—a Since / ——zdx Jx x—a is a non-oscillating function of x, it follows from Theorem 25 that L f ——dx xLaJ T x-a exists, and by Theorem 142 this limit must be infinite. 198 INFINITESIMAL ANALYSIS. Since 1/(^)1 >^ on the neighborhood under consideration, it follows from Theo- rem 107 and Corollary 2; Theorem 40, that L f f{x)d2 L x=a^ X exists and is infinite, which is contrary to the hypothesis. Theorem 143.! // (1) fii^) and /2(x) are of the same rank of infinity at x = a, or if /i(x) is of lower order than fiix), (2) / fi{x)dx and I f2ix)dx both exist for every x on the segment a b, (3) There is a neighborhood of x^a on which /2(x) does not change sign, P (4) L I f2(.x)dx is finite, J th£n it follows that L I f\{x)dx exists and is finite. t This is what Professor Moore in his lectures calls the relative con- vergence theorem. Theorems 143, 144, 151, 152 in this form are due to him. t We notice that since under the hypothesis /zCi) does not change sign, L I f2{x)dx L I f2{x)dx J X cannot fail to exist either finite or infinite, for it follows from this hypothesis that / fi{x)dx is a non-oscillating function of x and therefore, by Theorem 25 that the limit exists. IMPROPER DEFINITE INTEGRALS. 199 Proof. — Since from the hypothesis L I f2{x)dx exists and is finite, we have by Theorem 138 that for every t there exists a F*(a) such that for every xi and X2 on segment a b and on V*{a) IX X2 J2{x)dx xi <e. <M Consider Xi and X2 on a neighborhood of a; = a for which ^^ )/2(x) and for which /2(x) does not change sign. Then, by Theorem 113, I f''h(x)dx <M-\ f'^f2(x)d2 <M-c. Since M-e can be made small at will by making e small, it follows by Theorem 138 that £hix)d2 L TJJ iU X exists and is finite. An important special case of this theorem is when /i(x) is of the same or lower order of infinity than /2(x), i.e., L . , . =.K, a constant not zero. x=a/2(2;) The reader should verify for himself that Theorem 140 is a corollary of Theorem 143. The other previous tests for the existence of the improper definite integral can all be reduced to special cases of Theorem 143. Cf., for example, the logarithmic test on page 410 of Pierpont. Theorem 144. // (1) /i(x) and /2(x) are of the same rank of infinity at x=a, or if /i(x) is of higher order than f2{x), (2) / f\{x)dx and I f2{x)dx both exist I— I for every x on the segment a b, 200 INFINITESIMAL ANALYSIS (3) There is a neighborhood of x=a on which /i(x) does not change sign, (4) L I f2(x)dx is infinite (see note under Theorem 143), then L / fi (x)dx exists and is infinite or fails to exist.'f x= a\J X Proof. — This is a direct consequence of Theorem 143, since if L / /i(x)dx, x^aJ X which exists by the foot-note of Theorem 143, were finite, then L I f2{^)dx x'=aJ X would exist and be finite. Theorem 145. // for a function /i(x) which does not change sign in the neighborhood of x — a there exists a monotonic func- tion f2{x) infinite of the same rank as fi(x) as x approaches a, I fi{x)dx and / f2ix)dx both existing for every x on the seg- ment a b, then a necessary condition that L I fi {x)dx shall exist xLa\J X and be finite is that L (x-a)-/i(i)=0. Proof. — By hypothesis L rfx{x)dx t This is what Professor Moore calls the relative divergence theorem. IMPROPER DEFINITE INTEGRALS. 201 exists and is finite. Hence, by Theorem 143, x'=.aJ X exists and is finite. Therefore, by Theorem 142, L (x-a)-/2(x)=0. Since j^^ is bounded as x approaches a, i.e., |/i(2;)| <M- |/2(x)|, we have (i-a) ■ \h{x)\<M-{x-a) ■ \U{x)\. But L M.(x-a).|/2(a;)|=0. Therefore, by Corollary 4, Theorem 40, i'(x-a)-|/i(x)|=0, or by Corollary 2, Theorem 27, L ix-a)-fiix)=0. x~a § 2. The Definite Integral on an Infinite Interval. The integral over an infinite interval, viz.. L h{x)dx, c=oo«/ a has properties analogous to those of the improper definite inte- gral on a finite interval discussed in the preceding section, and is likewise called an improper definite integral. The following theorems correspond to Theorems 138 to 145. 202 INFINITESIMAL ANALYSIS. Theorem 146. // / f{x)dx exists for every x, a<x, then a necessary and sufficient condition that L f{x)dx exists and is finite, is that for every e there exists a D, such that for every two values of x, Xi and Xj, each greater than D„ I r'f(.x)d \x\<t Proof. — The theorem is a, direct consequence of Theorems 105 and 27. Theorem 147. // / f{x)dx exists for every x greater than a, and if L r\m\dx is finite,'\ then L f\x)dx exists and is finite. Proof. — The proof is like that of Theorem 139. Theorem 148. // / f{x)dz exists for every x greater than a, and if {x—a)^-f{x) is bounded as X approaches infinity for some k, k>\, then L rf{x)di exists and is finite. t Note on page 192 shows that this hypothesis is not redundant. IMPROPER DEFINITE INTEGRALS. 203 Proof. — ^If in the proof of Theorem 140 we write Z),i-* = ^ j^f instead of d.^-''= ' ~ \ and use Theorem 146 in- stead of 138, the proof of Theorem 140 will apply to Theorem 148. Theorem 149. // f{x) does not change sign for x greater than some fixed number D, and if for some positive number m and some number k^\ \{x-aY-f{x)\ >m for every x greater than D, then L f){x)dx cannot exist and be finite. Proof. — By making suitable changes in the proof of Theorem 141 so as to make xi and X2 approach infinity instead of a, that proof applies to this theorem. Theorem 150. // L / f{x)dx 1=00 »/(i exists and is finite, and if f{x) is monotonic for all values of x greater than some fixed number, then L (x-a)-/(x)=0. Proof. — By making slight modifications of the proof of Theorem 142, that proof applies to this theorem. Corollary. — If / f{x)dx exists for every x greater than a, and L f'f{x)dx « exists and is finite, and if /(x) does not change sign for x greater 204 INFINITESIMAL ANALYSIS. than some fixed number, then zero is a value approached by (x-a)/(x) as X approaches oo. The proof is similar to that of the corollary of Theorem 142. Theorem 151. // (1) fi{x) and /2(x) are infinitesimals of the same rank as x approaches 00, or if fi{x) is of higher order than f2{x), (2) / fi{x)dx and I f2{x)dx both exist for every x, a<x, (3) /2(2;) does not change sign for x greater than some fi^ed number, (4) L I f2{x)dx is finite, then it follows that L rfi{x)dx x^ootJa exists and is finite.^ Proof. — The proof is analogous to that of Theorem 143. Theorem 152. // (1) /i(x) and fzix) are infinitesimals of the same rank as x approaches infinity, or if fi{x) is of lower order than fiix), (2) / fi{x)dx and I f2{x)dx both exist for every x, a<x, %J a J a (3) /i(x) does not change sign for x greater than some fixed number, (4) L I f2{x)dx is infinite, then L I fi{x)dx exists and is infinite or fails to exist. Proof like that of Theorem 144. Theorem 153. // for a function fi{x) which does not change sign in the neighborhood of x=<xi there exists a monotonic func- tion fzix) such that /i(x) and /2(x) are infinitesimals of the same t See note under Theorem 143. IMPROPER DEFINITE INTEGRALS. 205 rank as X approaches infinity, I fi(x)dx and / f2(x)dxbotheJ>- isling for every x>a, then a necessary condition that L rh{x)dx shall exist and he finite is that L (x-a)-/i(x) = 0. The proof is like that of Theorem 145. § 3. Properties of the Simple Improper Definite Integral. The following definition of the simple improper definite integral is equivalent in substance to that given on page 192, and in form is partly the definition of the general improper definite integral given on page 210. The definite integral of a function is said to exist properly at a point Xi or in the neighborhood of this point, on the interval l-i I — I . . . . a 6 if there exists an interval on ai bi contaimngxi as anmterior point (or as an end point in case Xi=a or Xi=b) ^vrh that the proper definite integral of fix) exists on this : te val. The integral is said to exist improperly at a point xi on the interval aTb if f(x) has an infinite singularity at Xj and there exists an interval ai bi on a b containing Xi as an interior point (or end noint in case Xi=a or ii=6) such that the improper definite I — I . I— I integral exists on each of the intervals a, xi and xi 61. If on an interval a b the definite integral exists properly at every point except a finite number of points, and ex- ists improperly at each of these points, then the improper definite integral is said to exist simply on the interval iTb, or the simple improper definite integral is said to exist on 206 INFINITESIMAL ANALYSIS. 1-1 . I-I the interval a b. Let Xi, X2, . . . , Xn be the points of a 6 at which the integral exists improperly. The simple improper l-l definite integral on a 6 is the sum of the improper definite l-l I— I 1 1 I I — I integrals on the intervals a Xi, xi X2, ■ . . , x„_i x„, x„ b. We denote the simple improper definite integral of f{x) on l-l the interval o 6 by rmdx This symbol is used generically to include the proper as well as the improper definite integral. Theorem 154. // a<b<c, and if two of the three simple im- proper definite integrals I f{x)dx, I f(x)dx, and / f{x)dx S^a SJb S«/ o exist, then the third exists and rf{x)dx+ rf{x)dx^ rf{x)dx. 5«/ o St/ b St/ o Proof. — If 6 is a point at which the integral exists improperly, and if / f{x)dx and / f{x)dx sJa sJb S'Ja SJb both exist, then by the definition of rf(x)dx SJ a the latter exists and is equal to the sum of the two former. If one of the two integrals, say / Kx)dx, sJ a IMPROPER DEFINITE INTEGRALS. 207 exists, and if f" f{x)dx Sja exists, then / ^{x)dx • sjb exists since only in that case does frndx Sja exist. The equation / f{x)dx+ /fix)dx= f'f{x)dx Sja sJb sJ a likewise holds. If 6 is a point at which the integral exists properly, then the theorem follows from the above argument and the definition on page 205. Theorem 155. // / f{x)dx exists, then I f{x)dx sJb exists and I f{x)dx= — I f(x)dx. sJ a sJb Proof. — ^In case the integral exists improperly only at one point of the interval, then the theorem is an immediate consequence of Theorem 108 and CoroUary 1, Theorem 27. (If L fix) =K, then L{ —fix) j = —K.) The theorem in the general case follows directly from this case and the definition of the simple improper definite integral. Theorem 156. If c is a constant and if the simple improper 208 INFINITESIMAL ANALYSIS. definite integral of j{x) exists on a b, then the simple improper l-l definite iniegral of c-f{x) exists on a b and rf(x)dx=' rcf{x)dx. J a Sj a Proof. — ^The theorem is a direct consequence of Theorems 111 and 34. Theorem 157. // the simple improper definite integrals of fi{x) and fiix) both exist ona b, then the simple improper definite inte- gral of /i(x) +/2(x) and of /i(x) —f^ix) both exist and f\fl{x)±f2{x)\dx= rfl{x)dx± f)2{x)dx. SJa St/o Sja Proof. — ^The theorem is a direct consequence of Theorems 112 and 34. Theorem 158. // the simple improper definite integrals of /i(i) and fzix) both exist, and if fi{x) > fzix), then rf,{x)dx> rf2{x)dx. St/ a SJ a Proof. — The theorem is a direct consequence of Theorem 113 and Corollary 2, Theorem 40. Theorem 159. // / f{x)dx sJa exists, then I f{x)dx St/a is a continiums function of the limit of integration on the interval a 0. Proof. — If a; is a point at which the integral exists properly, the theorem is the same as 118. If a; is a point at which IMPROPER DEFINITE INTEGRALS. 209 the integral exists improperly, then the theorem follows from Theorems 138 and 27. Theorem i6o. // frndx sJa eocists, it does not follow that fimidx SJ a exists. Proof.— Let Xl, X2, X3, . . . , Xn, be an infinite sequence of points on 1 in the order indicated from 1 towards such that Jxn X n Consider a function /(i) defined as follows: fix) = - on xi 1, X3 xg, etc. 1 I— I I— I fw=-- on i2 a;i, X4 xs, etc. Obviously L I f(x)dx t exists and is finite since the series J— J + i . . . is convergent, while L C\f(x)\dx x=oJx is divergent since the harmonic series is divergent. t That is a limit point of the sequence of points is obvious since in case this sequence has a limit point greater than zero the proper definite integral of the function — would fail to exist on some interval ah where 0<a<6, X ' which is impossible. 210 INFINITESIMAL ANALYSIS. § 3. A More General Improper Integral. The problem of defining and studying the properties of the improper integral when the set of points of singularity is infinite has been treated by many writers.! In this section we give a few properties of improper integrals as defined by Harnack and Moore. I— I Denote by Po any set of points of content zero on a b, and by P the set of all points of a 6 not points of Po- P and Pq I— I are complementary sub-sets of a b. Denote by / any fimte set I— I of non-overlapping intervals of a 6 which contain no point of the set Pq. The symbol m(7) stands for the sum of the lengths of the intervals of I. Eor the sake of brevity D will be used for |a— 6|. The following conditions are assumed to be satisfied : (a) The definite integral of f{x) exists properly at every point of P. The sum of the integrals of /(x) on the intervals of / is denoted by Prndx. 'J a I (b) For every positive e there exists a positive 8. such that for any two sets, I' and I", of intervals none of which contain any point of Pq and for which |D-m(7')|<^. and \D-miI")\<8„ t A. Cauchy and B. Riemann studied the case of a fiflite number of sin- gularities in papers which are to be found in these writers' collected works. The infinite case has been treated by A. Hahnack, Mathematische Annalen, Vols. 21 and 24 (1883-84). O. Holder, Mathematische Annalen, Vol. 24 (1884). C. Jordan, Cours d' Analyse, Vol.' 2 (1894, 2d ed.). O. Stolz, Grundzuge der Differential- und Integralrechnung , Vol. 3. A. ScHOENFLiES, JahresbeHcht der Deutschen Mathematiker-Vereinigung Vol. 8 (1900). Valleb-Potjssin, LiouvUle's Journal, Ser. 4, Vol. 8 (1892). E. H. Moore, Transactions of the American Mathematical Society Vol 2 (1901). J. PlERPONT, Theory of Functions of Real Variables (1906). IMPROPER DEFINITE INTEGRALS. 211 / f{x)dx— I f{x)dx <e. \J al' JaV It follows by Theorem 27 that exists and is finite. This limit is denoted by r }{x)dx bJaP„ and is called the hroad improper definite integral with respect to Po of the fmiction /(x) on the interval a b. It is to be noticed that all the points of Po need not be on 1 — I . I— I a b] those which are not on a 6 do not affect the existence of / f{x)dx. W aPa Therefore if f{x) is improperly integrable on some sub-interval I — I I— I . . , , a' V of o h, its mtegral may be denoted by . r fix)dx. Wa'Pa Theorem i6i. If a<b<c and if of the integrals r f{x)dx, r f{x)dx, r fix)dx, WaPo b^bPo b-^ aPo either (a) / fix)dx and / fix)dx exist, <y (^) / f{x)dx exists, b'^aPo 212 INFINITESIMAL ANALYSIS, then all three integrals exist and /' fix)dx+ r f{x)dx= f f{x)dx. . . (1) JaPa -JlPo b^aPo I— I , , , Proof .—Every set I of intervals on o c may be regarded as composed of a set Ton a h and a set 7 on 6 c, while, conversely, every pair of sets / and / constitute a set I. Hence r Hx)dx= f f{x)dx+ I -f{x)dx. J al "J a'l '^ b I (Note that both members of this equation are multiple-valued functions of mil) and of m (7) and m(T)). The conclusion of our theorem follows in case (a) from Theorem 34. It remains to show that if / /(a;)dx exists, then / f{x)dx bJ a Po bJa Po and / j{x)dx exist, and in that case also equation (1) holds. bJ bPo Suppose that on some sequence of sets [/] one of the two expres- sions / _ f{x)dx and / _ f{x)dx, say / _ f{x)dx, approaches Jal J bl J al two distinct values as m(/) approaches D. Since there is some sequence of sets of intervals 1 7' | on which / _ /(x) approaches '^ bl only one value, it follows that on the sequence of sets of intervals obtained by associating with each 7 an V and with each 7' an 7', / f{x)dx approaches two distinct values as m{I) = D, which Jal is contrary to hypothesis. If / _f{x)dx approaches infinity, then clearly / f{x)dx must approach infinity of the opposite sign. Hence, by the corollary of Theorem 51a, / f{x)dx will approach both + oo mJ a I IMPROPER DEFINITE INTEGRALS. 213 and -00 as m{I)^D, which again contradicts the hypothesis that / f{x)dx exists. The equaUty I f{x)dx= f f{x)dx+ r f(x)dx b'J n P„ b'J a Po b'J b Pa now follows from the identity of the limitands / f{x)dx and / f{x)dx+ / _f{x)dx. 'J a I 'J a Y ^ bY Theorem 162. // / f{x)dx exists, then I f(x)dx exists W a. Pa b-J b P„ and f f(x)dx=- f f{x)dx. Wa Po b'Jb Po Proof.— By Theorem 108» for every / / f{x)dx= - I f{x)dx, J al «/ 6 / whence / ]{x)dx=- I j{x)dx. W aPo b^ b Po Theorem 163. // / f{x)dx exists, then I c-f{x)dx exists bJ a Pa b^ a Pa end I c-f(x)dx = c- I fix)dx. Wa Pa b'^a Po Proof. — This is a direct consequence of Theorems 111 and 34. Theorem 164. // / fi{x)dx and I f2{x)dx both exist, bJ a Po b^ o Pa then / (fi{x)±f2{x))dx exists and 214 INFINITESIMAL ANALYSIS. r hi^)dx± r J2{x)dx= f {h{x)dx±h{x))dx. W a Pa h^ a Po b^ a Pq Proof. — This is a direct consequence of Theorems 112 and 34. Theorem 165. // h{x) ^fzix), then r h{x)dx^ f f2{x)dx, provided these integrals exist. Proof. — By Theorems 113 and 40. Theorem 166. // / fi{x)dx and I hix)dx both exist, r hix)-f2{x)dx b-^aPo does not in general exist. Proof. — Let fi{x)=J2{x) = -j=. In this case the hypothesis of the theorem is verified but the product, - fails to be inte- I — I grable on the interval 1. Theorem 167. / f{x)dx is a continuous function of x. b^aPo Proof. — If a; is a point at which the integral exists properly, the continuity follows by Theorem 118. If a; is a point of the set Po, then, by Theorem 26, we need to show that for every e I — I there is a d, such that for every interval a' h' containing Xi and I n of length less than d„ \ I f{x)dx <s. By definition there |6>/o' Po exists a d, such that for every I' and /'' for which lm(/') -D\<d, md\mir)-D\<d., I r f(x)dx- f f{x)dx |«/ a P J a V <e. IMPROPER DEFINITE INTEGRALS. 215 I— I Let a' b' be an interval containing xi such that |a'-fe'l<y- Let /' be any set of intervals not containing any point of Pq and . . . I — I containing no point of a' h', and such that |m(/') - D| < d,. De- note hyl^a'b') any set of non-overlapping intervals on a' V con- taining no point of Pq, and let /" be the set of all intervals in /' and 7(„-i,o- Then \m(I")-D\<d. and /_ t{x)dx= /_ /(x)dx+ r_f(x)dx ^al" ^aV 'J a! Ka'V) \ pv (I Ph />b and / _ /(x)dd= / _ j{x)dx- / _/(x)da Hence / f{x)dx \b^ a' Po <e. Corollary. — For Xi any point on a 6 L x=xi b'^ Xj f'f{x)dx=0. Theorem i68. If f{x) is integrable with respect to Pq, and if Pi is a set of points of content zero, then f{x) is integrable with respect to the set P2 consisting of all points in Po and in Pi and r f{x)dx= f f{x)dx. Proof. — Obviously the set P2 is of content zero. Any set of intervals / not containing a point of P2 is also a set 1 not 216 INFINITESIMAL ANALYSIS. containing a point of Po- Hence any value approached by / J{x)dx as m(/) approaches D is a value approached by 'J al yf{x)dx as m(7) approaches D. Hence / f{x)dx exists and / f{x)dx= f f{x)dx. Theorem 169. If fi{x) is integrable vnth respect to Pi and J2{x) is integrable with respect to P2, then fi{x) ±f2{x) is integrable with respect to the set, P3, of all points in Pi and P2 and r f{x)dx± r f{x)dx= r (/iwi/acx))^. i«/ a Pi 6*^0 Pj 6«^ a Pa Proof. — By Theorem 168 each of the functions fi{x) and fzix) is integrable with respect to P3, and and / fi{x)dx= / fi{x)dx, WaPi WaP} / f2(x)dx= / f2{x)dx, Wa P, W a P, and hence, by Theorem 164, fi{x)±f2{x) is integrable with respect to P3 and r fi{x)dx+ r f{x)dx= r (j(x)±f{x))dx. W a Pi b^ a Pi W a Pi The broad improper definite integral as here defined con- tains as a special case the proper definite integral, the integral in that case existing properly at every point of the interval a h. It does not, however, contain as a special case the simple im- proper definite integral considered in § 3. This may readily be shown by means of the function used on page 209 to show IMPROPER DEFINITE INTEGRALS. 217 that the simple improper definite integral is not absolutely convergent. In the case of this function a sequence of sets of intervals /„ may be so chosen that / f{x)dx shall ap- proach any value whatever as m(/a) approaches D. An improper integral which includes both the simple and the broad improper integrals is obtained as follows; Every set / is to be such that if /' is its complementary set of seg- ments on a b, then every segment of I' contains at least one point of Pq. The limit of / f{x)dx as m(7) approaches D, if •-'a / existent, is called the narrow improper definite integral and is denoted by / f{x)dx. It is evident that if the broad integral exists, then the narrow integral also exists. The narrow integral includes the simple improper definite integral of the preceding chapter. Hence it follows that the broad and the narrow integrals are not equiva- lent.! Theorems 161 to 167 hold of the narrow integral as well as of the broad integral. The proofs are identical with the above except that the sets 7 are limited as in the definition of the narrow integral. It may be shown by examples that Theorems 168 and 169 do not hold in the case of the narrow integral. To show that 168 does not hold consider the function defined in the proof of Theorem 160, where Po consists of the point 0. Let Pi be the [xj of that example. Then obviously the narrow integral / ti^)dx, where P2 contains all the points of Pi and P2, fails to exist. The same example shows that Theorem 169 does not hold of the narrow integral. ■f The narrow integral is so called because it has fewer properties than the broad integral. It exi.sts for a wider class of functions. 218 INFINITESIMAL ANALYSIS. § 5. Special Theorems on the Criteria of Existence of the Improper Definite Integral on a Finite Interval. The examples of this section are intended to give an idea of the possible singularities of improperly integrable functions, and to indicate the difficulty of obtaining more general criteria of the divergence or convergence of the simple improper integral than those given in § § 1 and 2 of this chapter. Lemjna. — 'For every function /i(x) which is unboimded in every neighborhood of x=a there is a function /2(x) which is infinitesimal as x approaches a, such that ji{x)-f2{x) is un- bounded in every neighborhood of i = o, and such that /2(X) x—a is monotonic increasing as x approaches o. Proof. — Since /i(i) is imboimded in every neighborhood of x = o, it follows that for every point Xi of the segment a b there is a point X2 on the segment a xi such that |/ife)|>2|/i(xi)l>2M, and such that (xg —a) 5 J(xi — a). Let Xi, X2, X3, . . . , x„, . . . be a sequence of points dense only at a such that |/i(a;„)|>2|/i(x„_i)|>2''-i-M, and such that |x„— a| ^ i|x„_i -a\. We define /2(x) as follows: hM = - on the points Xi, X2, . . . , x„, . . . . TV IMPROPER DEFINITE INTEGRALS. 219 avd fzix) is linear between the points of the sequence Xi, X2, . . . , Xn,... Then there are values of x on a;„ x„_i such that l/i(a;)|-/2(x)>|-M, whence hi^)-hi^) is unbounded in the neighborhood of a.f Ubviously ^— ^ IS monotomc increasing as x approaches a. Theorem 170. For every function /i(x) which is unbounded in every neighborhood of x = a there exists a non-oscUlating func- tion f2{x) such that L I f2(x)dx exists and is finite, while (.x-a)-fi{x)-f2{x) is unbounded in the neighborhood of x=a. Proof. — According to the lemma there exists a function /3 (x) such that L fa (x) = 0, while fz{x)-fi(x) is imbounded and the fimction fsix) U{x) = x—a is monotonic increasing as x approaches a. Since ix-a)f4(x)-fiix)^f3ix)-f,ix), t In case L f,(x)=<x, f,(x) ==== or ftipc)^- — -^ would satisfy the / (x) requirements of the lemma except that they need not make -^^^ monotonic. 220 INFINITESIMAL ANALYSIS. (x-a)-}i{x)-fi{x) is unbounded in the neighborhood of x=a. I — I Let xi, . . . , a;„, . . . be a sequence of points on a b whose only- limit point is a, such that /3(x)-/i(x) is unbounded on this set. In the sequence (xi -o)/4(xi), (x2-a)/4(x2), ..., ix„-a)U{x)„ . (1) L (x„-o)/4(x„)=0, since L {x-a)fi{x)=0. 71=00 x=a Hence there is a value of n, rii, such that l(xi-a)/4(xi)l>2l(x„.-a)/4(xOI, and another value of n, Ui, such that |(x„,-a)/4(x„J| >2|(x„,-a)/4(x„,)|, etc., rim+i being so chosen that |(2„„-a)/4(x„JI > 2l(x„„^i -a)/4(a;n„+i)|. In this manner we select from the sequence (1) a set of tenns forming the convergent series (xi-a)/4(xi) + (x„, -a)/4(x„,) + . . . + (x„^-a)/4(x„J+. . . (2) We then obtain a function f^ix) as follows: For the set of values of X x„„^i<x<x„„, /2(X)=/4(X„„). Then (1) /zC^;) is non-oscillating since /4(a;"m)</4(a;nm+l). (2) {x-a)J2(x)-fi{x) is unbounded on the set ij, x^, Xr^, ■ ■ ■ , Xn„, ■■-, since on this set /2(X)=/4(X). IMPROPER DEFINITE INTEGRALS. 221 But the terms of this series are numerically smaller than the corresponding terms of the convergent series (2). Hence ^ / }2(x)dx exists and is finite. Theorem 170 may be regarded as showing that L (x-a)/2(a;)=0 is a strong necessary condition that, under the hypothesis of Theorem 142, ^ / f2{x)dx L z— a*/ X shall exist and be finite. For, according to Theorem 170, it is impossible to modify the function (a; -a) by any factor /i(i) which shall approach infinity so slowly that for every function fzix) where L / f2{x)dx \U X exists and is finite L(x-a)A(z)-/2(a:)=0.t x^a Theorem 171. For every function fiix) defirMi on the interval I — I a b there exists a function fzix) such that (1) fzix) is continuous and does not change sign on a certain neighborhood of x = a. t See Prinosheim, Mathematische Annilen, Vo!. 37, pp. 591-694 (1890). 222 INFINITESIMAL ANALYSIS. (2) L I f2(x)dx exists and is finite. (3) For X on a certain set [a/] x=a h{x') Proof. — Let Xi , x-i, . . . , x„', ... be a set of points of the I— I interval a b dense only at a. Let Bi, Bz, B3, . . . , Bn, ... be a set of numbers such that 5„-n|/i(x'„)|>2-fi„+i(n + l)i/i(z'„+i)|. (n = l, 2, 3, . . .) On the X axis lay off a set of segments [a„] such that a„ is of length Bn and a;„ is its middle point. On the segments (t„ as bases construct isosceles triangles on the positive side of the X axis whose altitudes are n-|/i(a;)|. The measures of areas of these triangles form a convergent series. Let fsix) be any continuous, monotonic, unbounded function such that L rf3(x)dx exists and is finite. We then define /2(a;) as the function repre- sented by the following curve : (1) Those parts of the boundaries of the isosceles triangles just described which lie above the curve defined by ]z{x). (2) Those parts of the curve defined by ]z{x) which he out- side the triangles or on their boundary. Obviously the func- tion so "defined has the properties specified in the theorem, the points xx, X2,..., Xn', . . . being the set [a/] specified by (3) of the theorem. Theorem 171 means that from the hypothesis that the improper definite integral of f{x) exists on a 6 it is impossible to obtain any conclusion whatever as to the order of infinity or the rank of infinity of f{x) atx=a. This is what one would IMPROPER DEFINITE INTEGRALS. 223 expect a -priori, since the definite integral is a function of two parameters, while the necessary condition in terms of t ounded- ness would be in terms of only one of these. § 6. Special Theorems on the Criteria of the Existence of the Improper Definite Integral on the Infinite Interval. Theorem 172. For every function fi{x) which is unbounded as X approaches 00 there exists a non-osciUating function /2(x) such that L I f2{x)dz z^oDc/a exists and is finite, while (x — a)fi{x)-f2^x) is unbounded as x approaches <x>. Proof. — Obviously the lemma of Theorem 170 can be stated so as to apply to the case where x approaches « instead of 0. If then in the proof of Theorem 161 the set of points ii . . . x„ . . . is so taken that L Xn=CO instead of a, the proof of Theorem 161 applies with the excep- tion that fzix) is non-oscillating decreasing instead of non-oscil- lating increasing. Theorem 173. For every function f\ {x) defined on the interval a 00 there exists a function /2(x) such that (1) /zCx) is corUinuou^ and does not change sign for x greater than a certain fixed number. (2) L I f2(x)dx exists and is finite. 224 INFINITESIMAL ANALYSIS. (3) For X on a certain set [x^ Proof. — Such a function /aCx) may be defined in a manner analogous to that of the proof of Theorem 171. The remarks as to the meaning of Theorems 170 and 171 apply with obvious modifications to Theorems 172 and 173. INDEX. 31 Absolute convergence series, 72 Absolute value, 14 Algebraic functions, 53 " numbers, 18 Approach to a limit, 60 Axioms of continuity, 4, ' ' of the real number system, 13 Bounds of indetermination, 84 " upper and lower, 3, 47 Broad improper definite integral, 211 Change of variable, 79, 126, 175 Class, 3 Closed set, 41 Constant, 44 Content of a set of points, 183 Contin ity at a point, 61 ' axioms of, 4, 31 " over an interval, 88 uniform, 89 Continuous function, 61 ' ' real number system, 4 Continuum, linear, 4 Convergence of infinite series, 71 to a limit, 60 Covering of interval or segment, 33 Decreasing function, 49 Dedekind cut, 7 Definite integral, 153 Dense, 41 Dense in itself, 41 Dependent variable, 44 Derivative, 117 " progressive and regres- sive, 118 Derived function, 120 Difference of irrational numbers, 8 Differential, 128 Differential coefficient, 130 Discontinuity, 62, 63 ' ' of the first and second kind, 85 The references are to pages, of infinite Discrete set, 41 Divergence, 71 Everywhere dense, 41 Exponential function, 54 Function, 44 " algebraic, 51 continuity of, at a point. 61 " continuity of, over an in- terval, 88 ' ' exponential, 54 " graph of, 46 " mfinite at a point, 101 " inverse, 45 " limit of, 60 " monotonic decreasing, 49 " " increasing, 49 " non-oscillating, 49 " oscillating, 49 " partitively monotonic, 50 " rational, 53 " " integral, 53 ' ' transcendental, 54 " unbounded, 47 " uniform continuity of, 89 ' ' upper and lower liound of, 47 ' ' value approached by, 60 Geometric series, 73 Grapn of a function, 46 Greatest lower bound, 4 Improper definite integral, 191 Improper definite integral, broad, 211 Improper definite integral, narrow, 217 Improper definite integral, simple, 205 Improper existence of the definite integral, 205 225 226 INDEX. Increasing function, 49 Independent variable, 44 Infinite, 101, 102 Infinite segment, 32 " series, 70 " " convergence and di- vergence of, 71 Infinitesimals, 75 Infinity as a limit, 40, 47, 60 Integral, definite, 153 existing properly at a point, 205 Integral oscillation, 155 Interval, 32 Inverse function, 45 Irrational number, 1, 4 ' ' numbers, difference of, 8 " product of, 8 " " quotient of, 9 " sum of, 8 Least upper bound, 4 L'Hospital's rule, 139 Limit, lower, 84 " of a function, 60 " of integration, 153 " point, 39 ' ' upper, 84 Limitand function, 60 Linear continuum, 4 Logarithms, 58 Lower bound of a function, 47 " " ■" " set of numbers, 3 " integral, 181 " segment, 12 Many-valued function, 44 Maximum of a function, 130 Mean-value theorem of the differen- tial calculus, 133 Mean-value theorem of the integral calculus, 169 Minimum of a function, 131 Monotonic function, 49 Narrow improper definite integral, 217 Necessary and sufficient condition, 65 Neighborhood, 38 Non-differentiable function, 150 Non-integrable function, 155 Non-numerably infinite set, 185 Non-oscillating function, 49 Nowhere dense, 41 Number, 1 " algebraic, 18 " irrational, 1, 4 Number, system, 4 Numbers, transcendental, 18 ' ' sequence of, 70 " sets of, 3 Numerably infinite set, 185 One-to-one correspondence, 30 Order of function, 102 Oscillation of a function, 49 " " " function at a point, 85 Partitively monotonic, 50 Perfect set, 41 Polynomial, 53 Product of irrational numbers, 8 Progressive derivative, 118 Proper existence of the definite inte- gral at a point, 205 Quotient of irrational mmibers, 9 Rank of infinitesimals and infinites, 114 Rational functions, 53 . ' ' integral functions, 53 " numbers, 1 Ratio test for convergence of infinite series, 73 Real mmiber system, 4, 13 Regressive derivative, 118 RoUe's theorem, 132 Segment, 32 infinite. 32 " lower, 12 Sequence of numbers, 70 Series, infinite, 70 " convergence and divergence of, 71 " geometric, 73 " Taylor's, 134, 135 Sets of numbers, 3 Simple improper definite integral, 205 Single-valued functions, 44 Singularity, 101 Sum of irrational numbers, 7 Taylor's series, 134 Theorem of uniformity, 35 Transcendental functions, 54 " numbers, 18 Unbounded function, 47 Uniform continuity, 89 Uniformity, 35 Upper bound of a function, 47 " "" set of numbers, 3 INDEX. 227 Upper integral, 181 " limit, 84 Value approached by a function, 60 «' " " the independ- ent variable, 60 Variable, 44 ' ' dependent, 44 " independent, 44 Vicinity, 38 V(a), 38 V*{a), 38 SHORT-TITLE CATALOGUE OF THE PUBLICATIONS or JOHN WILEY & SONS, New York. Lokdon: chapman & HALL, Ldhtbd. ARRANGED TTNDER SUBJECTS. 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Baker's Treatise on Masonry ConBtruction Svo, 5 00 Roads and Pavements Svo, s 00 Black's United States Public Works Oblong 4to, 5 00 ♦ Bovey's Strength of Materials and Theory of Structures Svo, 7 50 Burr's Elasticity and Resistance of the Materials of Engineering Svo, 7 50 Byrne's Highway Construction Svo, 5 00 Inspection of the Materials and Workmanship Employed in Construction. i6mo, 3 00 Church's Mechanics of Engineering. . Svo, 6 00 Du Bois's Mechanics-of Engineering. Vol. I Small 4to, 7 50 •Eckei's Cements, Limes, and Plasters Svo, 6 00 Johnson's Materials of Construction Large Svo, 6 00 Fowler's Ordinary Foundations Svo, 3 50 Graves's Forest Mensuration. Hvo, 4 co * Greene's Structural Mechanics. . Svo, 2 50 Keep's Cast Iron Svo, 2 50 Lanza's Applied Mechanics Svo, 7 50 Marten's Handbook on Testing Materials. (Henning.) 2 vols. Svo, 7 50 Maurer's Technical Mechanics. , . Svo, 4 00 Merrill's Stones for Building and Decoration Svo, 5 00 Merriman's Mechanics of Materials Svo, 5 00 * Strength of Materials i2mo, i 00 Metcalf 8 Steel. A Manual for Steel-users i2mo, 2 00 Patton's Practical Treatise on Foundations Svo, 5 00 Richardson's Modern Asphalt Pavements Svo, 3 00 Richey's Handbook for Superintendents of Construction i6mo, mor., 4 00 • Ries's Clays: Their Occurrence, Properties, and Uses Svo, 5 00 Rockwell's Roads and Pavements in France i2mo i 25 8 3 oo I 00 3 50 3 oo a oo 5 oo 8 oo 3 oo 3 so 3 so 4 oo Sabin's Industrial and Artistic Technology of Paints acd Varnish. 8vo, Smith's Materials of Machines i3mo, Snow's Principal Species of Wood 8vo, Spalding's Hydraulic Cement i3mo. Text-book on Roads and Pavements xsmo, Taylor and Thompson's Treatise on Concrete, Plain and Reinforced. 8vo, Thurston's Materials of Engineering. 3 Parts 8vo, Part I. Non-metallic Materials of Engineering and Metallurgy SyOp Part n. Iron and Steel gvo, Part m. A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8to, Tillson's Street Pavements and Paving Materials 8vo, Waddell's De Pontibus. (A Pocket-book for Bridge Engineers.). .i6mo, mor., 300 • Specifications for Steel Bridges i3mo, 50 Wood's (De V.) Treatise on the Resistance of Materials, and an Appendix on the Preservation of Timber 8vo, 3 00 Wood's (De V.) Elements of Analytical Mechanics 8vo, 3 00 Wood's (H. P.) Rustless Coatings; Corrosion and Electrolysis of Iron and SteeL 8vo, 4 00 RAILWAY ENGIWEERIHG. Andrew's Handbook for Street Railway Engineers 3x5 inches, morocco, x 35 Berg's Buildings and Structures of American Railroads 4to, 5 00 Brook's Handbook of Street Railroad Location. i6mo, morocco, i 50 Butf s Civil Engineer's Field-book. x6mo, morocco, 2 50 Crandall's Transition Curve x6mo, morocco, x 50 Railway and Other Earthwork Tables 8vo, i 50 Dawson's "Engineering" and Electric Traction Pocket-book . . i6mo, morocco, 5 00 Dredge's History of the Pennsylvania Railroad: (1879) Paper, 5 00 Fisher's Table of Cubic Yards Cardboard, 33 Godwin's Railroad Engineers' Field-book and Explorers' Guide. . .i6mo, mor., 3 so Hudson's Tables for Calculating the Cubic Contents of Excavations and Em- bankments 8vo, X 00 Molitor and Beard's Manual for Resident Engineers x6mo, x 00 Nagle's Field M<^n^*^ for Railroad Engineers x6mo, morocco, 3 00 Phitbrick's Field Manual for Engineers i6mo, morocco, 3 00 Searles's Field Engineering x6mo, morocco, 3 00 Railroad SpiraL x6mo, morocco, i 50 Taylor's Prismoidal Formuls and Earthwork 8vo, x 50 * Trautwine's Method of Calculating the Cube Contents of Excavations and Embankments by the Aid of Diagrams 8vo, 2 00 The Field Practice of Laying Out Circular Curves for Railroads. i3mo, morocco, 3 50 Cross-section Sheet Paper, 3S Webb's Railroad Construction x6mo, morocco, s 00 Economics of Railroad Construction Large xsmo, 3 50 Wellington's Economic Theory of the Location of Railways Small 8vo, S 00 DRAWHTG. 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Svo, 3 00 LAW. * Davis's Elements of Law Svo, 2 50 * Treatise on the Military Law of United States Svo, 7 oo * Sheep, 7 JO * Dudley's Military Law and the Procedure of Courts-martial . . . Large i2mo, 3 50 Mqni f l for Courts-martiaL i6mo, morocco, i 5® Wait's Engineering and Architectural Jurisprudence Svo, 6 oo Sheep, 6 so Law of Operations Preliminary to Construction in Engineering and Archi- tecture ...Svo 500 Sheep, 5 50 Law of Contracts. 8vo, 3 00 Winthrop's Abridgment of Military Law i2mo, 2 50 MAITOFACTURES. Bemadou'g Smokeless Powder— Nitro-cellulose and Theory of the Cellulose Molecule ""»»• Bolland's Iron Founder i2mo, The Iron Founder," Supplement 1 2mo, Encyclopedia of Founding and Dictionary of Foundry Terms Used in the Practice of Moulding i2mo, • Claassen's Beet-sugar Manufacture. (Hall and RoKe.) Svo, • Eckel's CemenU, Limes, and Plasters 8vo, Eissler's Modem High Explosives 8vo, Effront's Enzymes and their Applications. 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Ford's Boiler Making for Boiler Makers. iSmo, Hopkin's Oil-chemists' Handbook. 8™> Keep's Cast Iron. ^'O' 11 2 SO 2 50 2 SO 3 00 3 00 6 00 4 00 3 00 I 00 I 00 3 00 2 50 Leach's The Inspection and Aiutlysis of Food with Special Reference to State Control Large 8vo, 7 so * McKay and Larsen's Principles and Practice of Butter-making 8vo, i 50 Uatthewe's The Textile Fibres 8vo, 3 50 Metcalf's SteeL A Uanual for Steel-users: i2mo, 2 00 Hetcalfe'r Cost of Manufacttires — And the Administration of Workshops . 8to, 5 00 Heyer's Modem Locomotive Construction 4to, zo 00 Horse's Calculations used in Cane-sugar Factories i6mo, morocco, z 50 * Reisig's Guide to Piece-dyeing 8vo, 25 00 Rice's Concrete-block Manufacture 8vo, ^ 00 Sabin's Industrial and Artistic Technology of Paints and Vamislu 8vo, 3 00 Smith's PresE-workizig of Metals 8vo, 3 00 Spaldizig's HydrauUc Cement. z2mo, 2 00 Spencer's Hancfbook for Cheznists of Beet-sugar Houses. .... z6mo morocco* 3 00 Handbook for Cane Sugar Manufacturers i6mo morocco* 3 00 Taylor and Thoznpson's Treatise on Concrete, Plain and Reinforced 8vo, 5 00 Thurston's Manual of Steam-boilers, their Designs, Construction and Opera- tiozi. 8to, s do * Walke's Lectures on Explosives 8vo, 4 00 Ware's Beet-sugar Manufacture and Refining Small 8vo, 4 00 Weaver's Military Explosives 8vo, West's American Foundry Practice z2mo, Moulder's Text-book z2mo, Wolff's Windmill as a Prime Mover 8vo, 3 00 SO 50 3 Wood's Rustless Coatings: Corrosion and Electrolysis of Iron and Steel. .8vo, 4 00 MATHEMATICS. 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Berry's Temperature-entropy Diagram i2mo Carnot's Reflections on the Motive Power of Heat (Thurston.) i2mo, i so Dawson's "Engineering" and Electric Traction Pocket-book i6mo, mor., s 00 Ford's Boiler Making for Boiler Makers i8mo, Goss's Locomotive Sparks gvo Locomotive Performance gvo Hemenway's Indicator Practice and Steam-engine Economy . ismo 14 7 SO 6 00 2 50 6 00 2 SO 7 so 7 SO 4 00 5 00 I 00 2 00 3 00 1 00 8 00 3 50 2 so 2 00 3 00 »5 2 00 S 00 Button's Hechanical Engineering of Power Plants 8to, 5 00 Heat and Heat-engines 8to. s 00 Kent's Steam boiler Economy 8vo, 4 00 Kneass's Practice and Theory of the Injector 8to, i so HacCord's Slide-valves 8vo, 2 00 Meyer's Modem Locomotive Construction 4to, 10 00 Peabody's Manual of the Steam-engine Indicator lamo. i 50 Tables of the Properties of Saturated Steam and Other Vapors 8vo, i oo Thermodynamics of the Steam-engine and Other Heat-engines 8vo, 5 00 Valve-gears for Steam-engines 8vo, i 50 Peabody and Miller's Steam-boilers 8vo, 4 00 Pray's Twenty Years with the Indicator Large 8vo, 2 50 Pupin's Thermodynamics of Reversible Cycles in Gases and Saturated Vapors. 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Svo, 5 00 Wehrenfenning'sAnalysisandSofteningof Boiler Feed-water (Patterson) 8vo, 400 Weisbach's Heat, Steam, and Steam-engines. (Du Bois.) Svo, 5 00 Whitham's Steam-engine Design 8vo, s 00 Wood's Thermodynamics, Heat Motors, and Refrigerating Machines. . .8vo, 4 00 MECHANICS AND MACHINERY. Barr's Kinematics of Machinery Svo, 3 so * Bovey's Strength of Materials and Theory of Structures Svo, 7 50 Chase's The Art of Pattern-making i2mo, 2 50 Church's Mechanics of Engineering Svo, 6 00 Notes and Examples in Mechanics Svo, 2 00 Compton's First Lessons in Metal-working i2mo, i so Compton and De Groodt's The Speed Lathe i2mo, 1 ko Cromwell's Treatise on Toothed Gearing i2mo, i 50 Treatise on Belts and Pulleys i2mo, i 50 Dana's Text-book of Elementary Mechanics for Colleges and Schools. . i2mo, i 50 Dingey's Machinery Pattern Making i2mo, 2 00 Dredge's Record of the Transportation Exhibits Building of the World's Columbian Exposition of 1893 4*0 half morocco, s 00 Du Bob's Elementary Principles of Mechanics: VoL I. Kinematics 8vo, 3 so Vol n. Statics 8vo. 4 00 Mechanics of Engineering. VoL I SmaU4to, 7 SO VoL n Small 4to, 10 00 Durley's Kinematics of Machines 8™- 4 00 15 Fitzgerald's Boston Machinist i6mo, i oo Flather's Dynamometers, and the Measurement of Power iimo, 3 00 Rope Driving i2mo, 2 00 Goss's Locomotive Sparks 8vo, 2 00 Locomotive Performance 8vo, s 00 * Greene's Structural Mechanics 8vo, 2 so Hall's Car Lubrication i2mo, i 00 . Holly's Art of Saw Filins i8mo, 75 James's Kinematics of a Point and the Rational Mechanics of a Particle. Small 8va, 2 00 * Johnson's (W. W.) Theoretical Mechanics i2mo, 3 00 Johnson's (L. J.) Statics by Graphic and Algebraic Methods Svo, 2 00 Jones's Machine Design: Part I. Kinematics of Machinery Svo, i 50 Part n. Form, Strength, and Proportions of Parts Svo, 3 00 Kerr's Power and Power Transmission Svo, 2 00 Lanza's Applied Mechanics Svo, 7 50 Leonard's Machine Shop, Tools, and Methods Svo, 4 00 * Lorenz's Modem Refrigerating Machinery. (Pope, Haven, and Dean.). Svo, 400 MacCord's Kinematics; or, Practical Mechanism Svo, 5 00 Velocity Diagrams Svo, i 50 * Martin's Text Book on Mechanics, Vol. 1, Statics i2mo, 1 25 Maurer's Technical Mechanics. Svo, 4 00 Herriman's Mechanics of Materials Svo, s 00 * Elements of Mechanics i2mo, i 00 * Michie's Elements of Analytical Mechanics Svo, 4 00 *Parshall and Hobart's Electric Machine Design 4to, half morocco, 12 50 Reagan's Locomotives : Simple, Compound, and Electric. New Edition. Large i2mo, 3 00 Reid's Course in Mechanical Drawing Svo, 2 00 Text-book of Mechanical Drawing and Elementary Machine Design. Svo, 3 00 Richards's Compressed Air i2mo, i 50 Robinson's Principles of Mechanism Svo, 3 00 Ryan, Norris, and Hoxie's Electrical Machinery. VoL I Svo, 2 50 Sanborn's Mechanics: Problems Large 12310, i 50 Schwamb and Merrill's Elements of Mechanism 8to, 3 00 Sinclair's Locomotive-engine Running and Management. i2mo, 2 00 Smith's (O.) Press-working of Metals Svo, 3 00 Smith's (A. W.) Materials of Machines i2mo, i 00 Smith (A. W.) and Marx's Machine Design Svo, 3 00 Spangler, Greene, and Marshall's Elements of Steam-engineering Svo, 3 00 Thurston's Treatise on Friction and Lost Work in Machinery and Mill Work Svo, 3 00 Animal as a Machine and Prime Motor, and the Lawc of Energetics. i2mo, 1 00 Tillson's Complete Automobile Instructor i6mo, i 50 Morocco, 2 00 Warren's Elements of Machine Construction and Drawing Svo, 7 50 Weisbach's Kinematics and Power of Transmission. (Herrmann— Klein.). Svo, 500 Machinery of Transmission and Governors. (Herrmann — Klein.). Svo, 5 00 Wood's Elements of Analytical Mechanics Svo, 3 00 Principles of Elementary Mechanics i2mo, i 25 Turbines Svo, 2 50 The World's Columbian Exposition of 1893 4to, i 00 MEDICAL. De Fursac's Manual of Psychiatry. (RosanoS and Collins.) Large i2mo, 2 50 Ehrlich's Collected Studies on Immunity. (Bolduan.) Svo, 6 00 Hammarsten's Text-book on Physiological Chemistry. (Mandel.) 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Gold and Mercury 8vo, Goesel's Minerals and Metals: A Reference Book. i6mo, mor. * Iles's Lead-smelting izmo, Keep's Cast Iron 8vo, Kunhardt's Practice of Ore Dressing in Europe 8vo, Le Cbatelier's High-temperatuxe Measurements. (Boudouard — Burgess. )x2mo, MetcalTs SteeL A Manual for Steel-users. izmo. Miller's Cyanide Process. izmo, Minet's Production of Afaiminom and its Industrial Use. (Waldo.). . . . izmo, Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8to, Smith's Materials of Machines. izmo, Thurston's Materials of Engineering. In Three Parts 8vo, Part n. Iron and SteeL 8vo, Part HL A Treatise on Brasses, Bronzes, and Other Alloys and their Constituents 8™> Ulke's Modem Electrolytic Copper Refining 8yo, JimERALOGY. Barringer's Description of Minerals of Commercial Value. Oblong, morocco, 2 50 Boyd's Resources of Southwest Virginia 8vo, 3 00 Map of Southwest Virignia Pocket-book form. 2 00 Brush's Manual of Determinative Mineralogy. (Penfield.) 8vo, 4 00 Chester's Catalogue of Minerals 8". paper, i 00 Cloth, I zs Dictionary of the Hames of Minerals 8vo, 3 so Dana's System of Mineralogy Large 8vo. half leather. .2 so First Appendix to Dana's New " System of Mineralogy." Large 8vo, i 00 Text-book of Mineralogy 8vo, 4 00 Minerals and How to Study Them "•>«>• ' SO Catalogue of American LocaUties of Minerals Large Bvo, Manual of Mineralogy and Petrography >s™o Douglas's Untechnical Addresses on Technical Subjects "mo, i 00 Eakle's Mineral Tables ™' ' \^ Egleston's Catalogue of Minerals and Synonyms ■• ■ »»<>, so Goesel's Minerals and Metals: A Reference Book i6mo,mor. 300 Groth's Introduction to Chemical Crystallography (Marshall) "mo. i 25 17 7 SO 7 SO 3 00 2 SO 2 so 1 SO 3 00 2 oo I 00 2 so 4 oo I 00 8 00 3 50 z 50 3 00 00 IddingB's Rock Minerals 8vo, s oo HeiTill's Non-metallic Minerals: Their Occurrence and Uses 8to, 4 00 * Penfield's Notes on Determinative Mineralogy and Record of Mineral Tests. 8to, paper, 50 * Richards's Synopsis of Mineral Characters izmo, morocco, i 25 * Rles's Clays: Their Occurrence, Properties, and Uses 8vo, 5 00 Rosenbusch's Microscopical Physiography of the Rock- mak i ng Minerals. (Iddings.) 8vo, s 00 * Tillman's Text-book of Important Minerals and Rocks 8vo, 2 00 Boyd's Resources of Southwest Virginia 8vo, 3 00 Map of Southwest Virginia Pocket-book form 2 00 Douglas's Untecbnical Addresses on Technical Subjects i2mo, I 00 EisSler's Modem High Eiplosives "'' 4 "o Goeael's Minerals and Metals : A Reference Book . . i6mo, mor. 3 00 Goodyear's Coal-mines of the Western Coast of the United States i2mo, 2 50 Ihlseng's Manual of Mining 8vo, 5 00 • Iles's Lead-smelting i2mo, 2 so Kunhardt's Practice of Ore Dressing In Europe 8vo, i 50 Miller's Cyanide Process i2mo, i 00 O'Driscoll's Notes on the Treatment of Gold Ores 8vo, 2 00 Robine and Lenglen's Cyanide Industry. (Le Clerc.) 8vo, 4 00 • Walke's Lectures on Explosives 8vo, 4 00 Weaver's Military Explosives 8vo, 3 00 Wilson's Cyanide Processes i2mo, i 50 Chlorination Process i2mo, 1 50 Hydraulic and Placer Mining i2mo, 2 00 Treatise on Practical and Theoretical Mine Ventilation. i2mo, 125 SANITARY SCIENCE. Bashore's Sanitation of a Country House i2nio, 1 00 * Outlines of Practical Sanitation i2mo, i 25 FolweH's Sewerage. (Designing, Construction, and Maintenance. J 8vo, 3 00 Water-supply Engineering 8vo, 4 00 Fowler's Sewage Works Analyses I2m3, 2 00 Fuertes's Water and Public Health i2mo, x 50 Water-filtration Works i2mo, 2 50 Gerhard's Guide to Sanitary House-inspection x6mo, 1 00 Hazen's Filtration of Public Water-supplies 8vo, 3 00 Leach's The Inspection and Analysis of Food with Special Reference to State Control 8vo, 7 so Mason's Water-supply. (ConsideredprincipaUyfromaSanitaryStandpoint)8vo, 4 00 Examination of Water. (Chemical and Bacteriological) i2mo, i 25 * Merriman's Elements of Sanitary Engineering 8vo, 2 00 Ogden's Sewer Design i2mo, 2 00 Prescott and Winslow's Elements of Water Bacteriology, with Special Refer- ence to Sanitary Water Analysis i2mo, i 25 * Price's Handbook on Sanitation i2mo, 1 so Richards's Cost of Food. A Study in Dietaries i2mo, x 00 Cost of Living as Modified by Sanitary Science i2mo, x 00 Cost of Shelter .• i2mo, i 00 18 2 oo I so 4 oo S oo I oo 3 50 7 50 I 50 I oo Richards and Woodman's Air. Water, and Food from a Sanitary Stand- point 8vo, • Richards and Williams's The Dietary Computer 8vo. Rideal's Sewage and Bacterial Purification of Sewage 8vo, Tumeaure and Russell's Public Water-supplies 8vo, Von Bebring's Suppression of Tuberculosis. (Bolduan.) i2mo, Whipple's Microscopy of Drinking-water 8vo, Winton's Microscopy of Vegetable Foods 8vo, Woodhull's Notes on Military Hygiene i6mo. ♦ Personal Hygiene X2mo, MISCELLANEOUS. Emmons's Geological Guide-book of the Rocky Mountain Exctirsion of the International Congress of Geologists Large t-vo, i 5-. Ferrel's Popular Treatise en the Winds 8vo, 4 00 Gannett's Statistical Abstract of the World 24mOp 75 Haines's American Railway Management i2mo, 2 50 Ricketts's History of Rensselaer Polytechnic Institute 1824-1894. .Small 8vo, 3 00 Rotherham's Emphasized New Testament Large 8vo , 2 00 The World's Columbian I xposition of 1893 4to, i 00 Winslow's Elements of Applied Microscopy i2mo, x 50 HEBREW AND CHALDEE TEXT-BOOKS. Green's Elementary Hebrew Grammar i2mo, i 25 Hebrew Chrestomathy 8vo, 2 00 Gesenius's Hebrew and Chaldee Lexicon to the Old Testament Scriptures. (Tregelles.) Small 4to, half morocco, s 00 Letteris's Hebrew Bible 8vo, 2 as 19 oORNEaUNlVaSlTYllBfWRY slUL U 1991 MATHEMATlCSUBflARY ■¥• ^ AfS, '2?! ■