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  "body": "\n## Summary\n\nA real function on a closed interval is absolutely continuous when small total length of input intervals forces small total change of the output.\nIt is stronger than [[uniform continuity|0Q4DR37DVE]] and stronger than [[bounded variation|A8251BHCMP]].\nThe absolutely continuous functions are exactly the indefinite [[Lebesgue integrals|DWJF7ABPR1]].\nSo they are the functions for which the [[fundamental theorem of calculus|RX7JSHMP8Z]] holds in Lebesgue's form.\nThe analogous notion for measures is the [[absolutely continuous measure|P0P21KDV6E]].\n\n## Explanations\n\n### L1 human\n\nUniform continuity says that one short input interval gives a small change of the output.\nAbsolute continuity asks the same for many short intervals at once, as long as their total length is small.\nEvery [[Lipschitz function|0QNVQ4VD9T]], such as $\\sin x$, is absolutely continuous.\nThe [[Cantor function|NPBYYK63FE]] is continuous and increasing but not absolutely continuous: it rises from $0$ to $1$ on a set of total length $0$.\n",
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