Curve minus a point is affine from a rational function with poles only at a single point

Question

Suppose $C$ is a smooth projective curve over $k$, and let $p$ be a rational point. Using Riemann-Roch, it is easy to show that there exists a rational function that is regular everywhere except at $p$. I want to use this fact to show that $C\setminus p$ is affine, but I am not sure where to go.

I know there are a couple of proofs showing $C\setminus p$ is affine using other methods, but I want to show it using the existence of such rational function. Thanks in advance.

Answer 1

Given a non-constant rational function $f$ on $C$ with a pole only at $p$, this determines a nonconstant map $f:C\to\Bbb P^1$. As all maps of smooth projective curves are constant or finite, our map is finite, and hence affine. As $f^{-1}(\infty)=\{p\}$ as a set, this means $f^{-1}(\Bbb A^1)=C\setminus p$, and therefore $C\setminus p$ is affine.