Consider $f: \mathbb{C} \to \bar{\mathbb{C}}$, $z \mapsto \exp\left(\frac{1}{z}\right)$. Is it analytic in a neighborhood of $0$?
I feel like it should be (since $z \mapsto \frac{1}{z}$ is a chart, and $z \mapsto \exp(z)$ is analytic near $0$), but I lack confidence because I never seriously studied Riemann surfaces before.