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  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}$, let $f \\colon D \\to \\mathbb{R}$ be a function, and let $K \\ge 0$ be real.\nThe function $f$ is Lipschitz with constant $K$ if\n$$|f(x) - f(y)| \\le K |x - y| \\quad \\text{for all } x, y \\in D.$$\nIt is Lipschitz if it is Lipschitz with some constant $K$.\n\n## Remarks\n\n[[Lipschitz functions are uniformly continuous|0K1EWAR6AF]].\nA differentiable function on an interval with $|f'| \\le K$ is [[Lipschitz with constant $K$|0P64VE79X3]].\nFor example, $|x|$ and $\\sin x$ are Lipschitz with constant $1$. The function $\\sqrt{x}$ on $[0, 1]$ is not Lipschitz.\nA Lipschitz map with constant $K \u003c 1$ from a set to itself is a [[contraction|0CZRT49QNG]]. Mathlib defines it as `LipschitzWith`.\n\n## Explanations\n\n### L1 human\n\nThe slope of each chord of the graph of a Lipschitz function is at most $K$ in absolute value.\nSo the function never changes faster than a fixed rate.\n",
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