Preimage of a point by a non-constant harmonic function on $\mathbb{R}$ is unbounded

Question

Let $u$ be a non-constant harmonic function on $\mathbb{R}$. Show that $u^{-1}(c)$ is unbounded.

I am not getting what theorem or result to apply. Could anyone help me?

Answer 1

Let $u(x)=x$, for all $x \in \mathbb{R}$. Then $u''(x)=0$ for all $x$. But $u^{-1}(\{c\}) = \{c\}$. I think somebody is cheating you :-)