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  "body": "\n## Statement\n\nLet $U \\subseteq \\mathbb{R}^n$ be open, let $f \\colon U \\to \\mathbb{R}^m$ be [[differentiable|0135XZZQ8M]] on $U$, and let $x, y \\in U$ be points such that the segment $[x, y] = \\{x + t (y - x) : 0 \\le t \\le 1\\}$ lies in $U$.\nSuppose that $M \\ge 0$ and $\\|Df(z)\\| \\le M$ for each $z \\in [x, y]$, where $\\|Df(z)\\| = \\sup_{\\|h\\| \\le 1} \\|Df(z) h\\|$ is the operator norm.\nThen\n$$\\|f(y) - f(x)\\| \\le M \\, \\|y - x\\|.$$\n\n## Remarks\n\nFor $m = 1$, the inequality follows from the [[mean value theorem in several variables|03HGK7EW6R]]. For $m \\ge 2$ there is in general no equality form, but the inequality still holds.\nA standard proof applies the one-variable mean value theorem to $\\varphi(t) = (f(y) - f(x)) \\cdot f(x + t(y - x))$.\nOn a convex set $C \\subseteq U$ with $\\|Df\\| \\le M$ on $C$, the inequality holds for all $x, y \\in C$, so $f$ is [[Lipschitz on C|0TKG7T3K0E]].\n\n## Explanations\n\n### L1 human\n\nSuppose that the speed of change of $f$ is at most $M$ everywhere on a straight path. Then $f$ changes by at most $M$ times the length of the path.\nThis is like saying that a car that never goes faster than $100$ km/h cannot cover more than $200$ km in two hours.\n",
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