Uniform continuity on a set

Definition

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

Remarks

So $f$ is uniformly continuous on $S$ when its restriction to $S$ is uniformly continuous. A function can be uniformly continuous on a set but not on a larger set. For example, $x^2$ is uniformly continuous on $[0, 1]$ but not on $\mathbb{R}$. Mathlib defines it as UniformContinuousOn.

Explanations

L1 human

Uniform continuity on $S$ only looks at pairs of points inside $S$. Points outside $S$ play no role.