Let $(X,d_X)$ be a [metric space](/page/Metric%20Space), let $a \in X$, and let $f:X \to \mathbb{R}$ be a function, where $\mathbb{R}$ is equipped with the usual metric $d_{\mathbb{R}}(s,t)=|s-t|$. If $f$ is continuous at $a$, then there exist $r>0$ and $M \ge 0$ such that