Let $\overline{\Omega}$ be a bounded, closed and convex set of $\mathbb{R}^n$. Prove that $C^1(\overline{\Omega})$, the space of continuously differentiable functions on $\overline{\Omega}$, is not dense in the Hölder space $C^{0,\alpha}(\overline{\Omega})$, with $\alpha \in (0,1)$.
The problem has a hint, it says that if $f \in C^{0,\alpha}(\overline{\Omega})$, then extend it outside the compact set like $f(s(x))$ ($x \notin \overline{\Omega}$), where $s(x)=\{y: |x-y|=\operatorname{dist}(x,\overline{\Omega})\}$ (basically $f(x)$, for $x \notin \overline{\Omega}$, takes the value of the $f$ evaluated at the closest point $y \in \overline{\Omega}$ of $x$), and this is possible to define as $\overline{\Omega}$ is closed, convex. I do not see how to use the hint. Could you give me some insight on how to use this hint?