Open set and holomorphic functions

Question

Suppose $U$ is an open set in complex, and let $a\in U$. Let $f$ be a holomorphic function such that $f'(a)\neq 0.$ Will it guarantee that there exists a small ball centred at $a$ such that the derivative of $f$ is non-vanishing in that ball? I rather get the feeling that if it is continuously differentiable then that would be the case, but will this still be the case if $f$ is just holomorphic?

Answer 1

Yes, because a holomorphic function is automatically continuously differentiable. Indeed, the derivative of a holomorphic function is automatically again holomorphic, which means that every holomorphic function is actually infinitely differentiable.