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Let $f: X \to \mathbb{R}$ be a $C^{1,1}$ (that is $C^1$ with Lipschitz differential) function on a manifold $X$. Suppose that $f$ is smooth at all points of a subset $C \subset \text{Crit}f$ of critical points.

Does Sard's lemma hold in this case, that is, does the set $f(C)$ is of zero measure ? If it does, could someone give me a hint why ?

Thanks a lot

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  • $\begingroup$ Sard's lemma applies to $f$ restricted to any nbd $N$ of the critical set, so it is sufficient $f_{|N}\in C^k(N)$ $k\ge \dim(X)$ $\endgroup$
    – Pietro Majer
    Commented Aug 16, 2018 at 13:36
  • $\begingroup$ What do you mean by smooth on $C$? If you only know that $f\in C^{1,1}$, then Sard's theorem hold if $\dim X\leq 2$ otherwise there are counterexamples. If you clarify your question I can provide an answer. $\endgroup$
    – Piotr Hajlasz
    Commented Jan 11, 2019 at 21:50

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