Uniformly continuous map of metric spaces

Definition

Definition in terms of the metric

Suppose $(X, d_{X})$ and $(Y, d_{Y})$ are metric spaces (in other words, $X$ and $Y$ are sets and $d_{X}$ and $d_{Y}$ are metrics on $X$ and $Y$ respectively). A function $f : X \to Y$ is termed a uniformly continuous map if the following holds:

$\forall ε > 0 \exists δ > 0 : d_{X} (a, b) < δ ⟹ d_{Y} (f (a), f (b)) < ε$ .

Definition in terms of the uniform structure

Suppose $(X, d_{X})$ and $(Y, d_{Y})$ are metric spaces. A map $f : X \to Y$ is termed uniformly continuous if $f$ is a uniformly continuous map from $X$ to $Y$ with respect to the induced uniform structures on $X$ and $Y$ from their respective metrics.