I know that if $U$ is an open subset of the complex plane $\mathbb{C}$ and $f:U\to \mathbb{C}$ is a holomorphic function and $f$ is one-to-one, then the derivative $f'(z)$ is different from zero for every $z\in U$.
But how to prove converse of this problem:
If $f'(z)\neq 0, \;\forall z\in U$, then $f(z)$ is locally injective?