I have been asked to prove the following:
Have $g:[a,b]\rightarrow \mathbb{R}$ be continuous on $K = [a,b] \setminus \cup^\infty_{n=1} (\alpha_n,\beta_n)$. Then, for any $\epsilon > 0$, there is a $\delta > 0$ independent from $x,y$,
$$x \in K, y \in [a,b] : |x-y|< \delta \implies |g(x) - g(y)| < \epsilon$$
Any help would be greatly appreciated!