Recall that an automorphism of a Riemann surface is a bijective analytic map onto itself, and that the inverse map is then guaranteed to be analytic.

Let $\Delta$ denote the $\operatorname{disc}\{z \in \mathbb{C}|| z \mid<1\}$, and let $\Delta^{*}=\Delta-\{0\}$.

(a) Prove that an automorphism $\phi: \Delta \rightarrow \Delta$ with $\phi(0)=0$ is a Euclidian rotation.

[Hint: Apply the maximum modulus principle to the functions $\phi(z) / z$ and $\phi^{-1}(z) / z$.]

(b) Prove that a holomorphic map $\phi: \Delta^{*} \rightarrow \Delta$ extends to the entire disc, and use this to conclude that any automorphism of $\Delta^{*}$ is a Euclidean rotation.

[You may use the result stated in part (a).]

(c) Define an analytic map between Riemann surfaces. Show that a continuous map between Riemann surfaces, known to be analytic everywhere except perhaps at a single point $P$, is, in fact, analytic everywhere.