Let $H$ denote the upper half-plane and $F = \{z \in H:-1<\text{Re}(z)<1 \text{ and } |z| > 1 \}$ considered as a subset of the Riemann sphere. Also define $\widetilde{F} = F \cup \{z \in H: \text{Re}(z) = \pm 1 \}$ and $\widetilde{H} = \bigcup_{n \in \mathbb{Z}} \{\widetilde{F} + 2n \}$.
I am trying to show that there is a conformal map $f: F \rightarrow H$ so that $f$ extends continuously to a map $\overline{F} \rightarrow \overline{H}$ mapping $1 \mapsto 1$, $-1 \mapsto 0$, and $\infty \mapsto \infty$.
I next want to show that $f$ may be analytically continued to an analytic map $g: \widetilde{H} \rightarrow \mathbb{C} \setminus \{0,1 \}$ such that for every $z \in \widetilde{H}$, $g'(z) \neq 0$.