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Let $f$ be a function from $\mathbb R^n$ to $\mathbb R$, and $K = f^{-1}(0)$.

I want to know when $K$ has a null measure border. Since there exist (Cantor-like) closed set with empty interior of positive measure, then I have to put other hypotesis on $f$.

My question is: if $f$ is continuous, can I say that $K$ has a null measure border? And if the answer is no, what is the minimum integer $m$ (even infinite) such that the statement is true when $f\in C^m$?

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    If $K$ is any closed set whatsoever there exists a $C^\infty$ function $f$ with $K = f^{-1}(0)$, so the answer is no in general. – 2017-01-04
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    This theorem has a name? Or it is a consequence from the fact that the space is normal? – 2017-01-04
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    If $K$ is closed, then $f(x) = d(x,K)$ is continuous, and $f(x) = 0$ iff $x\in K.$ This works for metric spaces too. – 2017-01-04

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