$u$ harmonic show there exists $b\in\partial G$ such that $u(b)=u(a)$

Question

Let $u:G\rightarrow\mathbb{R}$ be harmonic and continuous on $\overline{G}$. Assume that $G$ is a bounded region and its boundary $\partial G$ is connected. Let $a\in G$, and show that there exists $b\in \partial G$ such that $u(b)=u(a)$.

I am not sure were to start. Any hints would be helpful.

Answer 1

Hint: The maximum and minimum values of $u$ over $\overline G$ occur on $\partial G,$ a connected set. This forces $u(\partial G)$ to be a particular set.