Strange thing about Weak Maximum\Minimum Principle?

Question

I feel confused about this problem. I think it is obvious using Weak Maximum\Minimum principles. Since for harmonic functions. If $\Omega$ is bounded and $u\in C^2(\Omega) \cap C^0(\overline \Omega)$, then $inf_{\partial \Omega} u \le u(x)\le sup_{\partial \Omega}u$. Since $u$ is continuous on $\partial \Omega$, so $u( \Omega)\subset u(\partial \Omega)$.

But the Weak Maximum\Minimum Principle didn't say anything about $\Omega$ connected. Where did I go wrong?

Can anyone offer me some help? Thanks so much!:D

Answer 1

You seem to infer, from continuity of $u$, that every number between $\inf_{\partial\Omega}u$ and $\sup_{\partial\Omega}u$ is in $u(\partial\Omega)$. That won't work without some connectedness assumption.