Let $f:\mathcal{M} \to \hat{\mathbb{C}}$ where $\mathcal{M}$ is a arbitrary Riemann surface and $f$ is a meromorphic function. Let $A \subset \mathcal{M}$. If $f:A \to B$ then $f:\partial A \to \partial B$.
Does this result hold? Does $A$ have to be compact? What's a nice concise but clear proof?
Thanks