Nonsmooth Morse-Sard theorems

TL;DR

This paper establishes a generalized Morse-Sard theorem for nonsmooth functions, showing that the images of certain critical sets with specific subdifferential properties are Lebesgue-null, extending classical smooth results to nonsmooth analysis.

Contribution

It proves a new nonsmooth Morse-Sard theorem for functions with Taylor expansions and subdifferentials, broadening the scope of critical value theorems beyond smooth functions.

Findings

The image of critical points with Taylor expansions of order n-1 is Lebesgue-null.

For lower semicontinuous functions in a02, the set of points with zero proximal subdifferential has null image.

The results extend classical Morse-Sard theorems to nonsmooth and subdifferential contexts.

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.