If $e^{f(z)}$ is constant can we say that $f(z)$ is constant?

Question

A quick question. If $e^{f(z)}$ is constant can we say that $f(z)$ is constant? ($z \in \mathbb{C}$). Can this be said directly without giving any proof or maybe stating a theorem? Thanks

Answer 1

You can say if if you assume f is continuous. Otherwise it's false.

Answer 2

\begin{align*} f(z) = \left \{ \begin{array}{cc} 0 \quad & \vert z \vert \leq 1 \\ 2 \pi \mathrm{i} \quad & \vert z \vert > 1 \end{array} \right. \end{align*}