a function constant on open ball implies constant everywhere

Question

let $U$ be a domain and $f:U \rightarrow \Bbb C$ be analytic function such that $f$ is constant on some open ball $D_r(z_0)$

how to prove $f$ must be constant on $U$

hints?

Answer 1

Show that the set where $f^{(k)}=0$ for every $k$ is both open and closed. Closed is clear; for open use power series.