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  "body": "\n## Definition\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \\colon X \\to Y$ be a function and let $K \\ge 0$ be a real number.\nThe map $f$ is Lipschitz with constant $K$ when\n$$d_Y\\bigl(f(x), f(y)\\bigr) \\le K \\, d_X(x, y) \\quad \\text{for all } x, y \\in X.$$\nThe map $f$ is Lipschitz when it is Lipschitz with some constant $K \\ge 0$.\n\n## Remarks\n\nThe smallest such $K$ is the Lipschitz constant of $f$.\nThe map $x \\mapsto |x|$ on $\\mathbb{R}$ is Lipschitz with constant $1$. The map $x \\mapsto \\sqrt{x}$ on $[0, \\infty)$ is not Lipschitz, because $\\sqrt{x} / x$ is not bounded near $0$.\nA Lipschitz map with constant $K \u003c 1$ is a [[contraction|0MT2S0ZX4Y]].\nA differentiable $f \\colon \\mathbb{R} \\to \\mathbb{R}$ with $|f'(x)| \\le K$ for all $x$ is Lipschitz with constant $K$, by the mean value theorem.\n\n## Explanations\n\n### L1 human\n\nA Lipschitz map cannot stretch distances by more than the factor $K$.\nFor a real function, this means that the slopes of all its chords are between $-K$ and $K$.\n",
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