Continuous map of metric spaces

In terms of the metric
Suppose $$(X,d_X)$$ and $$(Y,d_Y)$$ are fact about::metric spaces. 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.