Differentiable functions are continuous

Statement

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$, and let $a \in I$. If $f$ is differentiable at $a$, then $f$ is continuous at $a$.

Remarks

For $x \ne a$, write $f(x) - f(a) = \frac{f(x) - f(a)}{x - a} \cdot (x - a)$. The first factor tends to $f'(a)$ and the second to $0$. The converse fails. The absolute value is continuous but not differentiable at $0$. The Weierstrass function is continuous and nowhere differentiable. Mathlib states it as HasDerivAt.continuousAt.

Explanations

L1 human

If a graph has a tangent line at a point, it cannot jump there. A jump would make the slopes of the secant lines blow up.