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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\R$ be a [[monotone function|0WFF6AZPBP]].\nThen $f$ is [[differentiable|0F1WR9C83K]] at [[almost every|RARTMA0ZEC]] point of $[a, b]$.\nThat is, the set of points of $(a, b)$ where $f$ has no finite derivative is a [[null set|VQQM4Y550T]].\n\n## Remarks\n\nThe standard proof uses the [[Dini derivatives|36DXKAD2RZ]] of $f$ and the [[Vitali covering lemma|FCA0J1X07K]].\nFor nondecreasing $f$, one shows that each set $\\set{x : D^+ f(x) \u003e q \u003e p \u003e D_- f(x)}$ with rationals $p \u003c q$ is null.\nOne also shows that the set where $D^+ f(x) = \\infty$ is null.\nTogether with the same argument for the other pairs of Dini derivatives, this gives a finite derivative almost everywhere.\n\nFor nondecreasing $f$, the derivative $f'$ is moreover Lebesgue integrable, and $\\int_a^b f'(x) \\, dx \\le f(b) - f(a)$ for the [[Lebesgue integral|DWJF7ABPR1]].\nThe inequality can be strict: for the [[Cantor function|NPBYYK63FE]], the left side is $0$ and the right side is $1$.\nMonotonicity cannot be replaced by continuity: the [[Weierstrass function|01A0MK1730]] is continuous and differentiable at no point.\nThe theorem extends to functions of bounded variation, which are differences of monotone functions by the [[Jordan decomposition|KDT3E0VE0K]].\n\n## Explanations\n\n### L1 human\n\nA monotone function only goes up, or only goes down.\nIt can have jumps and corners, even infinitely many.\nThe theorem says that it is still differentiable at almost every point: the bad points take up no length at all.\n\n### L2 human\n\nThe theorem is the starting point of the theory of differentiation of real functions in measure theory.\nIt leads to the differentiation of indefinite Lebesgue integrals and to the fundamental theorem of calculus for absolutely continuous functions.\n[[Fubini's theorem on differentiation|XDEJCKRVQ4]] refines it for series of monotone functions.\nIn contrast with the Cantor function, absolutely continuous functions satisfy $\\int_a^b f' = f(b) - f(a)$.\n",
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