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  "body": "\n## Statement\n\nLet $n, m \\ge 1$, let $U \\subseteq \\R^n$ be an open set, and let $f \\colon U \\to \\R^m$ be [[Lipschitz continuous|0QNVQ4VD9T]].\nThen $f$ is [[Fréchet differentiable|08EVNYZF42]] at [[almost every|RARTMA0ZEC]] point of $U$ with respect to [[Lebesgue measure|2Z2QVWZ262]] on $\\R^n$.\n\n## Remarks\n\nFor $n = 1$ the theorem follows from Lebesgue's theorem: a Lipschitz function of one variable is [[absolutely continuous|ZT6F4C2YHK]], hence of bounded variation, hence differentiable almost everywhere.\nIn general one works with each coordinate of $f$, so one may take $m = 1$.\nApplying the case $n = 1$ on lines and Fubini's theorem, the directional derivative of $f$ in each fixed direction exists almost everywhere.\nOne then shows that almost everywhere it depends linearly on the direction, first for a countable dense set of directions, and the Lipschitz bound turns this into full differentiability.\nFor example, $f(x) = \\norm{x}$ is Lipschitz on $\\R^n$ and differentiable everywhere except at $0$.\nContinuity alone is not enough: the Weierstrass function is continuous on $\\R$ and differentiable nowhere.\n\n## Explanations\n\n### L1 human\n\nA Lipschitz function is one whose graph is never steeper than some fixed slope.\nSuch a function may have corners, but the theorem says the corners are rare: at almost every point the function has a tangent plane.\n\n### L2 human\n\nThe theorem is the starting point of analysis on rough sets and maps.\nIt lets one use the derivative of a Lipschitz map in integrals, for example in the area and coarea formulas of geometric measure theory.\nIt also shows that Lipschitz functions belong to the Sobolev space $W^{1,\\infty}$ locally, with weak gradient equal to the almost everywhere gradient.\nThe one-dimensional case is closely tied to [[absolute continuity|ZT6F4C2YHK]] and to [[Lebesgue's theorem on monotone functions|1Y4N1DDQNR]].\n",
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