A note on a corollary of Sard’s theorem

Abstract:

A corollary of Sard’s theorem is the following: Corollary. Let $f:K \to {R^n}$ be a smooth (i.e. $f \in {C^k},k \geq 1$) map defined on a compact subset $K$ of ${R^n}$. Let $C = \{ y|{f^{ - 1}}(y)\;is\;infinite\}$. Then the Lebesgue measure of $C$ is zero. The purpose of this note is to show that a similar version of this theorem holds for Lipschitz functions.