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arXiv · 1605.01513

Nonsmooth Morse-Sard theorems

Daniel Azagra·Juan Ferrera·Javier Gomez-Gil
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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.

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Daniel Azagra, Juan Ferrera, Javier Gomez-Gil. 2016-05-05. Nonsmooth Morse-Sard theorems. https://arxiv.org/abs/1605.01513

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