My favorite characterization of continuity of a single-variable function is that a function $\displaystyle f:(D\subseteq \mathbb{R}) \to \mathbb{R}$ is continuous over its entire domain if and only if:

$\displaystyle \sup_{a\in D}(\inf_{\delta>0}(\sup_{0< |h|<\delta}{|f(a+h)-f(a)|}))=0.$