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  "body": "\n## Statement\n\nLet $I$ be an interval, let $f \\colon I \\to \\mathbb{R}$ be continuous and differentiable at each interior point of $I$, and let $K \\ge 0$.\nIf $|f'(x)| \\le K$ for each interior point $x$ of $I$, then $f$ is [[Lipschitz|0QNVQ4VD9T]] with constant $K$.\n\n## Remarks\n\nThe [[mean value theorem|0BARQY52N5]] gives $|f(y) - f(x)| = |f'(c)| \\, |y - x| \\le K |y - x|$.\nFor example, $\\sin$ is Lipschitz with constant $1$ on $\\mathbb{R}$.\nMathlib states a form for convex sets as `Convex.lipschitzOnWith_of_nnnorm_deriv_le`.\n\n## Explanations\n\n### L1 human\n\nIf the slope of the graph never exceeds $K$ in size, no chord can be steeper than $K$.\n",
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