Why is $f$ constant in this case? Reference: Conway - Functions of one complex variable p.55 Exercise 17.
Let $\epsilon>0$.
Let $f:B(0,\epsilon)\rightarrow \mathbb{C}$ be an analytic function such that $|f(z)|=|f(w)|$ for each $z,w\in B(0,\epsilon)$.
In this case, how do I prove that $f$ is constant?
I'm completely lost where to start..

Suppose that $f$ is not constant.  Then $f'$ is not identically zero.  Suppose, for instance (and without loss of generality), that $f'(0)\neq 0$ and $f(0)=1$.  Now use the definition of the derivative to estimate $f(z)$ when $z$ is very close to $0$.  Can you show there must be some $z$ such that $|f(z)|\neq 1$?
