If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then is it a constant?

Question

If a function $f: \mathbb{C}\to\mathbb{C}$ is bounded, then it is a constant. Is it true or false?

Answer 1

It is false. Let $f(z)=0$ for all $z\ne0$, and $f(0)=1$.

Now, if you require $f$ to be analytic, then the assertion is true: it's Liouville's Theorem, as mentioned by julien.

Answer 2

If you don't require that $f$ is analytic then this is false, as Martin Argerami says, but you can find a continuous function which is bounded and not constant:

$$f(z)=\begin{cases} z & |z|\leq1 \\\\ \frac z{|z|} & \text{otherwise}\end{cases}$$