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For a continuous function $u: \mathbf{R^2}\to \mathbf{R}$, consider the zero level set $E:=\{x\in \mathbf{R^2}:u(x)=0\}$, write $Int(E)$ as the collection of the interior points of the set $E$, my question is then, if $L^2(E\setminus Int(E))$ equals to zero or not. By $L^2$ I mean the 2 dimensional Lebesgue measure.

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Depending on $u$ it may or may not have zero measure. To have nonzero, take $E \subset \mathbb{R}^2$ to be a closed set of positive measure without interior points (for example a fat Cantor set) and $u(x) := \textrm{dist}(x,E)$.

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