What is the meaning of Uniformly_continuous?

That for every real ε > 0 there exists a real δ > 0 such that for all pairs of points x and y in X for which $D_{X}(x,y)<\delta$ , it must be the case that $D_{Y}(f(x),f(y))<\epsilon$ (where D_X and D_Y are the metrics of X and Y, respectively).