I'm attempting to solve a question which says that $ f: U \rightarrow \mathbb{C}$ is holomorphic on an open subset $U \subset \mathbb{C}$ and that $f'(a) \neq 0$ for some $a \in U$.
I'm asked to show that $f$ is injective on some disk around $a$ - I thought I had a proof but see now I only showed that $f(z) \neq f(a)$ for all $z$ in the disk with $z \neq a$, which obviously isn't good enough.
The section of the textbook I'm using that this exercise is in has covered the argument principle, Rouche's theorem, the open mapping theorem, and the inverse function theorem - so I assume that I'm supposed to use one of those to do so, but I can't see how.
Your help would be very much appreciated!