If M and N are compact Riemann surfaces, show that : $ f: M \to N $ biholomorphic mapping if and only if there is a finite set A, B, where $A \subset M, B\subset N$, such that $$ g: M-A \to N-B$$ is a biholomorphic mapping.
Here I have a similar problem: If M is a compact Riemann surface, and $ M-\{p\} \cong \mathbf{C}$, where p is a point in M, then $ M \cong \mathbf{C} \cup \{\infty\}$.
I have tried many times through definition. However, I still can't think clearly and give an isomorphic mapping.
Hint: use local holomorphic charts and Riemann's removable singularities theorem.
Edit: here are a little more details.
Let $f: M \backslash \{p\} \to \mathbb C$ be an isomorphism. Let $\phi: U \to \mathbb C$ be a local holomorphic chart of $M$ near $p$. Let $g(z)=\frac{1}{f(\phi^{-1}(z))}$, and $V:=\phi(U)$, $z_0:=\phi(p)$. Then $g: V \backslash \{z_0\} \to \mathbb C$ is an injective holomorphic map. Can you then prove:
that up to restricting $U$ if necessary, $g(V \backslash \{z_0\})$ is bounded?
that $g$ extends holomorphically to $V$?
conclude.
This is for the second part of your question, which is a special case of the first. The more general method is the same, you just need to replace the chart $z \mapsto \frac{1}{z}$ by local charts for $N$.