Differentiability at a Point Implies Continuity There

Let $\mathbb{R}$ be the set of real numbers, regarded as a metric space through the metric $d_{\mathbb{R}}$ of The Absolute Value Metric on the Real Line.

Let $p,q\in\mathbb{R}$, let $g:(p,q)\to\mathbb{R}$ be a function on the open interval $(p,q)$, and let $c\in(p,q)$.

If $g$ is differentiable at $c$, then $g$ is continuous at $c$ relative to $(p,q)$, as a map from $(p,q)$ into the metric space $(\mathbb{R},d_{\mathbb{R}})$.