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  "body": "\n## Summary\n\nA real function on a closed interval has bounded variation when the total up-and-down movement of its values is finite.\nThis total movement is the [[total variation|8QA5DEWNYD]] of the function.\nFunctions of bounded variation are exactly the differences of two [[monotone functions|0WFF6AZPBP]], by the [[Jordan decomposition|KDT3E0VE0K]].\nThey are the natural integrators for the [[Riemann–Stieltjes integral|JT9AFHG0NA]] and they are differentiable almost everywhere.\n\n## Explanations\n\n### L1 human\n\nWalk along the graph of a function from left to right and count how far the value goes up and down in total.\nIf this distance is finite, the function has bounded variation.\nEvery monotone function on $[a, b]$ has bounded variation, since its values move in one direction only.\nThe function $x \\sin(1/x)$ on $[0, 1]$, with value $0$ at $0$, does not.\nIt swings up and down infinitely often near $0$, and the sizes of the swings add up to infinity.\n",
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