Uniform continuity

Definition

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

Remarks

Each uniformly continuous function is continuous. In the epsilon-delta form of continuity, $\delta$ may depend on the point. Here one $\delta$ works for all points. The function $x^2$ is continuous on $\mathbb{R}$ but not uniformly continuous. Mathlib defines it for uniform spaces, as UniformContinuous. Eduard Heine published the first definition of uniform continuity, in 1870.

Explanations

L1 human

For a uniformly continuous function, one input tolerance works everywhere in the domain. The graph has no places where it becomes steeper and steeper without bound.