The Zygmund Morse-Sard Theorem

The classical Morse-Sard Theorem says that the set of critical values off:R ^n+k →R ⁿ has Lebesgue measure zero iff ∈C ^k+1. We show theC ^k+1 smoothness requirement can be weakened toC ^k+Zygmund. This is corollary to the following theorem: For integersn >m >r > 0, lets = (n ?r)/(m ?r); iff:R ⁿ →R ^m belongs to the Lipschitz class Λ _s andE is a set of rankr forf, thenf(E) has measure zero.