This is exercise $3.5$ from Stichtenoth's Algebraic Function Fields and Codes:
Let $F/K$ be a function field in one variable and $\mathbb{P}_F$ the set of its places. If $S\subsetneq \mathbb{P}_F$ is such that $\mathbb{P}_F\setminus S$ is finite, show that $\mathcal{O}_S=K[x_1,...,x_r]$ for some $x_1,...,x_r\in\mathcal{O}_S$
[here $\mathcal{O}_S$ denotes the holomorphy ring $\bigcap_{P\in S}\mathcal{O}_P$].
I think I'm able to see this is true in the special case when: $$F=K(x)$$ $$\mathbb{P}_{K(x)}\setminus S=\{P_{p_1(x)},...,P_{p_r(x)}\}$$
where $P_{p_i(x)}$ is the place associated to the irreducible polynomial $p_i(x)\in K[x]$. It's easy to see that $\mathcal{O}_S$ is the ring where the only poles allowed are $P_{p_i(x)}$ of any order, so:
$$\mathcal{O}_S=K\left[\frac{1}{p_1(x)},...,\frac{1}{p_r(x)}\right]$$
For the general case, I'm totally lost. Any suggestions?