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So my question is the following:

Let f be a real Lipschitz continuous function defined on a an interval of R.

Consider the set of points that are zeros of the function and every neighborhood of the point contains a non zero of the function.

Does this set have a null measure?

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closed as off-topic by Pietro Majer, Yoav Kallus, András Bátkai, Pace Nielsen, Kostya_I Jul 1 at 12:14

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No. Consider $f(x) = \operatorname{dist}(x, E)$ where $E$ is a "fat Cantor set".

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