Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. Let $f:X \to Y$ be a map. Suppose there exists a constant $c \ge 0$ such that $d_Y(f(x),f(y)) \le c\,d_X(x,y)$ for all $x,y \in X$. Then $f$ is uniformly continuous. In particular, if $f:X \to X$ is a [contraction mapping](/page/Contraction%20Mapping) on a [metric space](/page/Metric%20Space) $(X,d_X)$, then $f$ is uniformly continuous.