I can't seem to find an example of a function $f \colon \mathbb{R}^2\to \mathbb{R}^2$ which is $C^1$ and such that the set of its critical values is not of zero measure.
What examples are there?
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This has been known for some time, including the higher-dimensional problem, in $\mathbb{R}^n$, that if $f\in C^k$ where $k<n$ then the set of critical points need not be of zero measure. H. Whitney, A function not constant on a connected set of its critical points, Duke Math. J. 1 (1935), 514-517. |
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My favourite example is as follows. Let the simple curve $\kappa:[0,1]\to K\subset \mathbb{R}^2$ be a parametrization of (half of) the Koch curve, and let $\phi:K\to[0,1]$ be its inverse; it is a continuous function, and, due to the fact that $\kappa$ has infinite variation on any non-empty interval $J\subset [0,1]$, it satisfies $$|\phi(x)-\phi(y)|=o(|x-y|)$$ uniformly on $K$. Therefore the data $\phi$ together with the zero field on $K$ satisfy the hypotheses of the Whitney extension theorem for the case of $C^1$ regularity. Thus $\phi$ extends to a $C^1$ function $f:\mathbb{R}^2\to\mathbb{R}$ whose gradient vanishes identically on $K$. |
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