Uniformly continuous map of metric spaces

Definition

Definition with symbols

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:

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