#### Proof If $R(x)=\frac{p(x)}{q(x)}$ is a rational function, then $p(x)$ and $q(x)$ are polynomials and the domain of $R$ is $\{x \mid q(x) \neq 0\} .$ The [[Limit of a Quotient]] states that for all $c$ in the domain of a rational function, $ \lim _{x \rightarrow c} R(x)=R(c) $ So a rational function is continuous at every number in its domain.