Let $D \subseteq \mathbb{C}$ be a domain, let $(f, U)$ be a function element in $D$, and let $\alpha:[0,1] \rightarrow D$ be a path with $\alpha(0) \in U$. Define what it means for a function element $(g, V)$ to be an analytic continuation of $(f, U)$ along $\alpha$.

Suppose that $\beta:[0,1] \rightarrow D$ is a path homotopic to $\alpha$ and that $(h, V)$ is an analytic continuation of $(f, U)$ along $\beta$. Suppose, furthermore, that $(f, U)$ can be analytically continued along any path in $D$. Stating carefully any theorems that you use, prove that $g(\alpha(1))=h(\beta(1))$.

Give an example of a function element $(f, U)$ that can be analytically continued to every point of $\mathbb{C}_{*}$ and a pair of homotopic paths $\alpha, \beta$ in $\mathbb{C}_{*}$ starting in $U$ such that the analytic continuations of $(f, U)$ along $\alpha$ and $\beta$ take different values at $\alpha(1)=\beta(1)$.