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  "body": "\n## Statement\n\nLet $a \u003c b$ be real numbers and let $f \\colon [a, b] \\to \\R$ be a function.\nThen $f$ is [[absolutely continuous|ZT6F4C2YHK]] on $[a, b]$ if and only if the following three conditions hold:\n\n1. $f$ is [[continuous|0P27W9RW0G]] on $[a, b]$;\n2. $f$ has [[bounded variation|A8251BHCMP]] on $[a, b]$;\n3. $f$ has the [[Luzin N property|WMHMGJD261]].\n\n## Remarks\n\nAn absolutely continuous function is continuous and of bounded variation, and covering a null set by small open intervals shows that it has the N property.\nFor the converse, one shows that if $f$ is differentiable at every point of a measurable set $E$, then the outer measure of $f(E)$ is at most $\\int_E \\abs{f'}$.\nThe N property handles the null set where $f'$ fails to exist, and this controls the changes of $f$ over families of short intervals.\nNo one condition can be dropped.\nThe [[Cantor function|NPBYYK63FE]] is continuous and monotone but fails the N property.\nThe function $x \\sin(1/x)$ on $[0, 1]$ is continuous with the N property, being smooth away from $0$, but it does not have bounded variation.\nA step function is monotone and maps every set to a finite set, but it is not continuous.\n\n## Explanations\n\n### L1 human\n\nThe theorem splits absolute continuity into three simpler conditions.\nThe function must not jump, must not wiggle infinitely much, and must not spread a set of length zero over a set of positive length.\n\n### L2 human\n\nThe theorem explains exactly what goes wrong with the [[Cantor function|NPBYYK63FE]] and other [[singular functions|P5AWRD2793]]: they are continuous and monotone, so only the N property can fail.\nIt is often used to prove absolute continuity of compositions and inverses.\nFor example, a continuous strictly increasing function is absolutely continuous exactly when it maps null sets to null sets.\n",
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