Define the terms Riemann surface, holomorphic map between Riemann surfaces and biholomorphic map.

Show, without using the notion of degree, that a non-constant holomorphic map between compact connected Riemann surfaces must be surjective.

Let $\phi$ be a biholomorphic map of the punctured unit disc $\Delta^{*}=\{0<|z|<1\} \subset \mathbb{C}$ onto itself. Show that $\phi$ extends to a biholomorphic map of the open unit disc $\Delta$ to itself such that $\phi(0)=0$.

Suppose that $f: R \rightarrow S$ is a continuous holomorphic map between Riemann surfaces and $f$ is holomorphic on $R \backslash\{p\}$, where $p$ is a point in $R$. Show that $f$ is then holomorphic on all of $R$.

[The Open Mapping Theorem may be used without proof if clearly stated.]