Let $C$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points of $C$.

We say that $f$ is totally ramified at a point $p$ if the ramification index of $p$ equals $\deg(f)$.

Does $C$ admit a finite map $f \colon C \to \mathbb P^1$ which is totally ramified at $x$ and $y$?