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 $C(\overline \Omega)\subset H_0^1 (\Omega)\cap C(\Omega)$?

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 $\overline\Omega$?

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 $\overline\Omega$?

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 $C(\overline \Omega)\subset H_0^1 (\Omega)\cap C(\Omega)$?

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

Thank you any and all in advance.

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 $C(\overline \Omega)\subset H_0^1 (\Omega)\cap C(\Omega)$?

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 $\overline\Omega$?