Category of Riemann surface is balanced?

Question

Let $A$ and $B$ are Riemann surfaces. $f:A→ B$ is morphism of Rieman surface, in other word, holomorhic function, and supposed that $f$ is bijective. Then, can we say that inverse of $f$ is also holomorphic?

Thank you in advance.

Answer 1

"Yes".

In suitable coordinates, a holomorphic mapping is locally represented by a convergent power series $z \mapsto w = a_{n}z^{n} + \cdots$ with $n \geq 1$ and $a_{n} \neq 0$. Such a mapping is locally bijective if and only if $n = 1$, if and only if it locally has a holomorphic inverse.