Continuous map of metric spaces

Definition

In terms of the metric

Suppose $(X,d_{X})$ and $(Y,d_{Y})$ are Metric space (?)s. In other words, $X$ and $Y$ are sets, and $d_{X}$ and $d_{Y}$ are metrics on these sets. A function $f:X\to Y$ is termed a continuous map from $X$ to $Y$ if it satisfies the following:

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

In terms of the induced topology

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

The notion of continuous map of metric spaces gives rise to the notion of the category of metric spaces with continuous maps.