I was working on holomorphic functions and Riemann surfaces, and I was wondering about the inverse of a constant function:
Let $f:U\rightarrow V$ be a holomorphic function between two Riemann surfaces $U$ and $V$. If $f$ is constant, with $f(u_0)=v_0$, what happens with the inverse $f^{-1}:V\rightarrow U$? Is it even defined in $v_0 \in V$? And if it's defined there, is the following true: $f^{-1}(v_0)=U$?
Any help with clarification would be appreciated.