Let $U$ be a bounded, open subset of $\mathbb{R}^n$, and suppose that the boundary $\partial U$ is of class $C^1$. Suppose that $u \in W^{1, \infty}(U)$.
I wish to prove that there exists a continuous representative $u^*$ of $u$. That is, I want to show the existence of $u^* \in C^0(U)$ such that $u^* = u$ almost everywhere.
This is part of Theorem 5 in section 5.6 from Evans' book Partial Differential Equations. Evans proves that elements of $W^{1,p}(U)$ for $n < p < \infty$ actually have Hölder continuous representatives, but he leaves the $p = \infty$ case to the reader.
I'm not sure how to start with this $u \in W^{1, \infty}(U)$ and produce a $u^* \in C^0(U)$. My main conceptual difficultly is that $u$, as an element of $W^{1, \infty}(U)$ is not even defined pointwise. So I'm not sure how I would be able to produce a function $u^*$ which is defined pointwise. Evans' proof when $n < p < \infty$ utilizes the density of $C^\infty_0(\mathbb{R}^n)$ in $W^{1, p}(\mathbb{R}^n)$, but in the $p = \infty$ case we do not have such density results at our disposal.
I'm also not sure if $\partial U$ needs to be $C^1$ in the $p = \infty$ case.
Hints or solutions are greatly appreciated.
