Uniformly continuous map

Let \((X, d_X)\) and \((Y, d_Y)\) be metric spaces. A map \(f\colon X\to Y\) is uniformly continuous if for every \(\varepsilon > 0\) there exists \(\delta > 0\) such that for all \(x_1, x_2\in X\), \[ d_X(x_1, x_2) < \delta \implies d_Y(f(x_1), f(x_2)) < \varepsilon. \]