If the zeros of a holomorphic function $f$ have an accumulation point, then $f$ is constant.

Question

I'm having trouble with the proof of the next statement:

Let $U\subset \mathbb{C}$ be open and connected. If $f$ is defined in $U$ and is holomorphic, and the set of its zeros have and accumulation point, then $f$ is constant on $U$.

I know how to prove this when that accumulation point is in $U$, but I don't know what to do in the other case. Hope you can help me, please.

Thank you.

Answer 1

It's not true. For example, $\sin(1/z)$ with $U = \mathbb C \backslash \{0\}$.