Some remarks about the Morse–Sard theorem and approximate differentiability

Abstract

Let n, m be positive integers, $$n\ge m$$. 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 $$f:\mathbb {R}^n\rightarrow \mathbb {R}^m$$ 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 $$\mathbb {R}^m$$).