Nonsmooth Morse-Sard theorems

Abstract

We prove that every function $f:{\mathbb{R}}^n\to \mathbb{R}$ satisfies that the image of the set of critical points at which the function $f$ has Taylor expansions of order $n-1$ and non-empty subdifferentials of order $n$ is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential ${\partial }_P$, we see that for every lower semicontinuous function $f:{\mathbb{R}}^2\to \mathbb{R}$ the set $f(\{x\in {\mathbb{R}}^2:0\in {\partial }_{P}f(x)\})$ is ${\mathcal{L}}^1$-null.

arXiv commentFinal version. The main result has been strengthened thanks to the suggestions of a referee