Suppose $\Omega$ is a bounded domain in $\mathbb R^3$ with Lipchitz boundary $\partial\Omega$, and $u\in H\_0^1(\Omega)\cap C(\Omega)$. Is $u$ continuous to the boundary i.e. do we have $u \in C( \overline{\Omega})$?


In other words, is is true that $H\_0^1 (\Omega)\cap C(\Omega)\subset C(\overline \Omega)$?


Depending on the answer(s), I may have some follow-up questions (for what it's worth).


Thank you any and all in advance.


**Edit**: It seems the answer is no so I am adding follow-up questions: Can I get that $u$ is bounded and/or attains its maximum on $\overline\Omega$?



Is an $H_0^1$ function continuous to the boundary if it is continuous in the interior?
Not necessarily- let $\Omega = B\_1 \cap \{x\_3 > 0\}.$ Then $u(x) := (1-|x|^2)\frac{x\_3}{|x|}$ is in $H^1\_0(\Omega) \cap C^{\infty}(\Omega),$
but $u$ is discontinuous at the origin.


