# Some remarks about The Morse-Sard theorem and approximate differentiability

## TL;DR

This paper explores the connection between approximate differentiability of higher order functions and the Morse-Sard property, establishing conditions under which the critical set's image has measure zero.

## Contribution

It demonstrates that locally Lipschitz functions with higher order approximate differentiability almost everywhere satisfy the Morse-Sard property under certain measure conditions.

## Key findings

- Functions with higher order approximate differentiability have measure-zero critical value sets.
- The Morse-Sard property holds for functions with specific differentiability and measure conditions.
- The paper clarifies the relationship between differentiability and measure-theoretic properties of critical sets.

## Abstract

Let $n, m$ be positive integers, $n\geq m$. We make several remarks on the relationship between approximate differentiability of higher order and Morse-Sard properties. For instance, among other things we show that if a function $f:\mathbb{R}^n\to\mathbb{R}^m$ is locally Lipschitz and is approximately differentiable of order $i$ almost everywhere with respect to the Hausdorff measure $\mathcal{H}^{i+m-2}$, for every $i=2, \dots, n-m+1$, then $f$ has the Morse-Sard property (that is to say, the image of the critical set of $f$ is null with respect to the Lebesgue measure in $\mathbb{R}^m$).