The Behavior of a Function on Its Critical Set

Referring to the next section for any unfamiliar notation or definition, let us consider the following statement. If m > 0 n > 1, R is an open subset of En , and f is a function on R to El of class Cm then f transforms its critical set into a set of (linear) measure zero. H. Whitney2 has shown the statement false if m = n 1 > 1; M. Morse and A. Sard, in an unpublished paper, have shown, on the other hand, that it is true providing m is the greatest integer in n + (n 3)2/16. This last result contains as a special case the fact that the statement is true in case m = n = 1, 2, 3, 4, 5, 6. Thus it is natural to ask: Is the statement true in case m =n? Theorem 4.3 of the present paper answers this question in the affirmative. As a matter of fact the answer is yes under the less restrictive assumption that f be of class C(m1-) providing each partial derivative of order (m 1) is totally differentiable. We give here no proof of this last statement; of the two proofs with which we are familiar one involves a fairly comprehensive modification of Section 3 as well as an additional lemma while the other involves some properties of an (n 1) dimensional measure of sets contained in E. . An obvious corollary of Theorem 4.3 is Theorem 4.4 which states that a function on an open subset of En to El which is of class C' is constant on any connected subset of its critical set. Theorems 4.3 and 4.4 are fairly direct consequences of Theorem 4.1 which is itself closely related to Theorem 3.7. Theorems 3.7 and 4.1 really form the kernel of the paper. We believe Theorem 4.1 in particular is susceptible of wider application than herein given.