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