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  "body": "\n## Definition\n\nLet $S$ be a subset of $\\mathbb{R}^n$, let $f : S \\to \\mathbb{R}^m$ be a [[function|0EPAT51DXQ]], and let $K \\ge 0$ be a real number.\nThe function $f$ satisfies a Lipschitz condition on $S$ with constant $K$, or is $K$-Lipschitz on $S$, if\n$$\\| f(x) - f(y) \\| \\le K\\, \\| x - y \\| \\quad \\text{for all } x, y \\in S.$$\nLet $D \\subseteq \\mathbb{R} \\times \\mathbb{R}^n$ and $f : D \\to \\mathbb{R}^m$. The function $f$ is Lipschitz in $x$ on $D$ with constant $K$ if $\\| f(t, x) - f(t, y) \\| \\le K \\| x - y \\|$ whenever $(t, x), (t, y) \\in D$.\n\n## Remarks\n\nA $K$-Lipschitz function is uniformly continuous.\nA differentiable function on a convex set with $\\| f'(x) \\| \\le K$ at each point is $K$-Lipschitz, by the mean value inequality. For example, $\\sin$ is $1$-Lipschitz on $\\mathbb{R}$.\nThe function $x \\mapsto \\sqrt{|x|}$ is continuous but not Lipschitz on any interval that contains $0$. The [[non-uniqueness example|00K7ASZ1PR]] uses a function of this kind.\nThe [[Picard–Lindelöf theorem|0P6KRD5B7A]] needs a Lipschitz condition in $x$.\n\n## Explanations\n\n### L1 human\n\n\nA Lipschitz function cannot change faster than a fixed rate.\nThe slope of each chord of its graph is at most $K$ in absolute value.\n",
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