THE ZYGMUND MORSE-SARD THEOREM

The classical Morse-Sard Theorem says that the set of critical values of f : R(n+k) --> R(n) has Lebesgue measure zero if f is-an-element-of C(k+1). We show the C(k+1) smoothness requirement can be weakened to C(k+Zygmund). This is corollary to the following theorem: For integers n > m > r > 0, let s = (n - r)/(m - r); if f : R(n) --> R(m) belongs to the Lipschitz class LAMBDA(s) and E is a set of rank r for f, then f(E) has measure zero.