Epsilon-delta definition of continuity

Statement

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a \in D$. Then $f$ is continuous at $a$ if and only if the following holds. For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - f(a)| < \varepsilon$ for all $x \in D$ with $|x - a| < \delta$.

Remarks

Each open interval around a point contains an interval of the form $(c - r, c + r)$ with $r > 0$. This gives the equivalence. Many texts take this condition as the definition of continuity. Mathlib states a form for metric spaces as Metric.continuousAt_iff.

Explanations

L1 human

For each allowed error $\varepsilon$ in the output, there is a tolerance $\delta$ in the input. Inputs within $\delta$ of $a$ give outputs within $\varepsilon$ of $f(a)$.