Uniformly continuous map of metric spaces

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 defining ingredient::uniformly continuous map from $$X$$ to $$Y$$ with respect to the induced uniform structures on $$X$$ and $$Y$$ from their respective metrics.