Examples of Riemann-Roch application

Question

The RR Theorem is often quoted as a means to constrain rational functions on curves. I would like to see easy examples of application of this theorem for this purpose on the Riemann Sphere $\mathbb{P}^1(\mathbb{C})$. For example we can consider the sheaf $\mathcal{O}_{\mathbb{P}^1}(D)(U)=\left\{f \in K(\mathbb{P}^1)^{\times}\mid \text{div}(f)|_U+D|_U \geq 0\right\}$ for any divisor $D$ and $U \subset \mathbb{P}^1$. I know that $\mathbb{P}^1(\mathbb{C})$ has genus $0$ so RR claims something like $\text{dim}\,H^0(\mathbb{P}^1,D) - \text{dim}\,H^1(\mathbb{P}^1,D) = 1+\text{deg}(D)$ and we pick $D$ to be $0 - \infty$ it becomes $\text{dim}\,H^0(\mathbb{P}^1,D) - \text{dim}\,H^1(\mathbb{P}^1,D) = 1$. How does this constrain the rational functions on an open subset $U$ in practice? Any other examples would be greatly appreciated too.