Uniformly continuous map

A map between metric spaces where one delta works uniformly for all points for a given epsilon.

Uniformly continuous map

A uniformly continuous map between metric spaces $(X,d_X)$ and $(Y,d_Y)$ is a map $f\colon X\to Y$ such that for every $\varepsilon>0$ there exists $\delta>0$ with

d_X(x,y)<\delta \implies d_Y\bigl(f(x),f(y)\bigr)<\varepsilon \quad\text{for all } x,y\in X.

Uniform continuity strengthens continuity by requiring $\delta$ to depend only on $\varepsilon$ (not on the point of $X$ ). It is implied by Lipschitz continuity , and it ensures that Cauchy sequences are sent to Cauchy sequences.

Examples:

The map $f(x)=\sin x$ from $\mathbb{R}$ to $\mathbb{R}$ is uniformly continuous.
The map $f(x)=x^2$ on $\mathbb{R}$ is not uniformly continuous, but it is uniformly continuous on any bounded interval such as $[0,1]$ .