Let $\Omega \subseteq \mathbb{R}^N$ be open and bounded, and set:
$d(x):=\text{dist} (x,\partial \Omega) =\inf_{y\in \partial \Omega} |x-y|\;.$ I would appreciate if somebody could verify my proof. I tried to show that it is Lipschitz continuous:
Let $\forall x,y \in \Omega$, and WLOG assume that $d(x)\geq d(y)$. Let $\forall \varepsilon >0$. By definition of infimum, $\exists z \in \partial \Omega$ such that $d(y)+\varepsilon > |y-z|$ and $|x-z| \geq d(x)$. Putting everything together, we obtain that $0 \leq d(x)-d(y) \leq |x-z| - |y-z| +\varepsilon \leq |x-y| + \varepsilon $. Since $\varepsilon$ was arbitrary, done.