(i) Let $f(z)=\sum_{n=0}^{\infty} a_{n} z^{n}$ be a power series with radius of convergence $r$ in $(0, \infty)$. Show that there is at least one point $a$ on the circle $C=\{z \in \mathbb{C}:|z|=r\}$ which is a singular point of $f$, that is, there is no direct analytic continuation of $f$ in any neighbourhood of $a$.

(ii) Let $X$ and $Y$ be connected Riemann surfaces. Define the space $\mathcal{G}$ of germs of function elements of $X$ into $Y$. Define the natural topology on $\mathcal{G}$ and the natural $\operatorname{map} \pi: \mathcal{G} \rightarrow X$. [You may assume without proof that the topology on $\mathcal{G}$ is Hausdorff.] Show that $\pi$ is continuous. Define the natural complex structure on $\mathcal{G}$ which makes it into a Riemann surface. Finally, show that there is a bijection between the connected components of $\mathcal{G}$ and the complete holomorphic functions of $X$ into $Y$.