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23 SARD'S THEOREM 169

each s such that E_m n I(s, a/N) is nonempty, if x₀ is a point of this set, it
follows from (16.23.1.3) that for every other point x e E_m n I(s, <z/N) we have
||f(x) - f(x₀)|| g M(a/N)^m+1, and therefore

A(f(E_m n I(s, a/N))) ^ M
where A is Lebesgue measure on R^p. Hence
A(f(E_M n I(k, a))) g MV⁽

Since by hypothesis m ^ w/p, the right-hand side of this inequality tends to
zero with 1/N, and the proof is complete.

Sard's theorem implies, in particular:

(16.23.2) Let X, Y be pure differential manifolds of dimensions n,p, respec-
tively, such that n <p, and let f: X ->• Y be a C°°-mapping. Then Y -/(X) is
dense in Y.

In other words, for C°°-mappings there do not exist phenomena of the
type of the "Peano curve" (Section 4.2, Problem 5, or Section 9.12, Prob-
lem 5).

PROBLEMS

1. (a) Let m, n, p, r be integers >0. Let f be a C°°-mapping of an open set U <=• R^m into
R", and let g be a C°°-mapping of an open set V => f(U) in R" into R^p. Put h = g o f
Show that for each x e U we have

D'h(x) = E 2 o-_r(i,,..., /,)D«g(f (x)) o (D¹>f(x), ..., D'f (x)),

fl»0 (li.....lq)

where in the inner sum, (/i,..., /„) runs through all sequences of q integers ^1 such
that /!+••• + /, = r. Furthermore, in this formula, the constants cr_r(/i, ...,/<,) are
rational numbers which depend only on the //, q, m, n, p, and r, and not on the functions
f and g.

(b) Let x₀ e U. Assume now only that, for some s < r, f is a mapping of class C^r~^s of
U into R", and that g is a mapping of class C^r of V ^ f(U) into R^p. Suppose also that
D^fcg(f(x₀)) = 0 for k <J s. For each x e U and each integer A: e [0, r], let

h.W = Z Z v_k('\ ,..., /«)D«g(f(x)) o (D^Jlf(x),..., D^f"f(x)) e ^_fc(R^w; R"),

« = S+ 1 (ll.....iq)

the second sum being over the same sequences (i_t,..., /_a) as in (a), so that // <£ k — s for
all /, and hence the function h_fc is well-defined on U. By applying Taylor's formula,
show that h_k(x) can be written in the form



	23 SARD'S THEOREM 169 each s such that E_m n I(s, a/N) is nonempty, if x₀ is a point of this set, it follows from (16.23.1.3) that for every other point x e E_m n I(s, <z/N) we have \|\|f(x) - f(x₀)\|\| g M(a/N)^m+1, and therefore A(f(E_m n I(s, a/N))) ^ M where A is Lebesgue measure on *R^p.* Hence A(f(E_M n I(k, a))) g MV⁽ Since by hypothesis m ^ w/p, the right-hand side of this inequality tends to zero with 1/N, and the proof is complete. Sard's theorem implies, in particular: (16.23.2) Let X, Y be pure differential manifolds of dimensions n,p, respec- tively, such that n <p, and let f: X ->• Y be a C°°-mapping. Then Y -/(X) is dense in Y. In other words, for C°°-mappings there do not exist phenomena of the type of the "Peano curve" (Section 4.2, Problem 5, or Section 9.12, Prob- lem 5).

	PROBLEMS 1. (a) Let m, n, p, r be integers >0. Let f be a C°°-mapping of an open set U <=• R^m into R", and let g be a C°°-mapping of an open set V => f(U) in R" into R^p. Put h = g o f Show that for each x e U we have D'h(x) = E 2 o-_r(i,,..., /,)D«g(f (x)) o (D¹>f(x), ..., D'f (x)), *fl»0 (li.....lq)* where in the inner sum, (/i,..., /„) runs through all sequences of q integers ^1 such that /!+••• + /, = r. Furthermore, in this formula, the constants cr_r(/i, ...,/<,) are rational numbers which depend only on the //, q, m, n, p, and r, and not on the functions f and g. (b) Let x₀ e U. Assume now only that, for some s < r, f is a mapping of class C^r~^s of U into R", and that g is a mapping of class C^r of V ^ f(U) into R^p. Suppose also that D^fcg(f(x₀)) = 0 for k <J s. For each x e U and each integer A: e [0, r], let h.W = Z Z v_k('\ ,..., /«)D«g(f(x)) o (D^Jlf(x),..., D^f"f(x)) e ^_fc(R^w; R"), *« = S+ 1 (ll.....iq)* the second sum being over the same sequences (i_t,..., /_a) as in (a), so that // <£ k — s for all /, and hence the function h_fc is well-defined on U. By applying Taylor's formula, show that h_k(x) can be written in the form