I can't seem to find an example of a function $f \colon \mathbb{R}^2\to \mathbb{R}$ which is $C^1$ and such that the set of its critical values is not of zero measure.
What examples are there? $\phantom{aaa}$
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I edited the question to its original form as a questions about functions tom $R^2$ to $R$.
Piotr Hajlasz
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A counter example for Sard's theorem in the case C^1
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