Let $(X,d)$ be a [metric space](/page/Metric%20Space) containing at least two distinct points, and let $f:X\to X$ be a map. Then $f$ is a contraction if and only if there exists a constant $c\in[0,1)$ such that, with the supremum taken in $[0,\infty]$,
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\begin{align*}
\sup\left\{\frac{d(f(x),f(y))}{d(x,y)}:x,y\in X,\ x\ne y\right\}\le c.
\end{align*}