Continuous representative of functions in ${H_{0}}^1(\Omega)$

Question

Let $\Omega \subset \mathbb{R}^2$ be an open and bounded Lipschitz domain. Does every function in $H_0^1(\Omega)$ have a continuous representative? I know that not every function in $H^1(\Omega)$ has a continuous representative and I wonder why this should hold true for $H_0^1(\Omega)$. However, that's used in a proof.

Answer 1

No, I don't think so. Let us consider the function $u=(\log\log\frac{4R}{|x|})\chi$, where $\chi$ is a smooth cut-off function such that $\chi(x)=1$ if $|x|<\frac{R}{2}$ and $\chi(x)=0$ for all $|x|>R$. This function belongs to $H_0^1(B_{2R})$ but it is not even in $L^\infty(B_{2R})$. Obviously, it doesn't have any continuous representation.

The proof for this result can be found in Adams' "Sobolev space" page 110.