Differentiability at an Interior Point Implies Continuity There

Let $\mathbb{R}$ be the real numbers, and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ equipped with the metric determined by the absolute value.

Let $I\subseteq\mathbb{R}$ be an interval, let $f:I\to\mathbb{R}$, and let $x_0\in I$ be an interior point of $I$ at which $f$ is differentiable.

Then $f$ is continuous at $x_0$ relative to $I$, regarded as a map from the subset $I$ of $(\mathbb{R},d_{\mathbb{R}})$ into $(\mathbb{R},d_{\mathbb{R}})$.