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:

$\forall \ \varepsilon >0\ \exists \delta >0:d_{X}(a,b)<\delta \implies d_{Y}(f(a),f(b))<\varepsilon$.

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.