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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f \\colon [a, b] \\to \\R$ be a function.\nThe *total variation* of $f$ on $[a, b]$ is\n$$V_a^b(f) = \\sup_P \\sum_{i=1}^{n} \\abs{f(x_i) - f(x_{i-1})},$$\nwhere the [[supremum|0BTMQ04556]] runs over all [[partitions|069P20KYQT]] $P$ of $[a, b]$ given by $a = x_0 \u003c x_1 \u003c \\cdots \u003c x_n = b$.\nThe value $V_a^b(f)$ lies in $[0, \\infty]$.\n\n## Remarks\n\nExample: for $f(x) = \\sin x$ on $[0, 2\\pi]$, the total variation is $4$.\nThe function is monotone on $[0, \\pi/2]$, $[\\pi/2, 3\\pi/2]$ and $[3\\pi/2, 2\\pi]$, and on each of these pieces the variation is the absolute change $1$, $2$ and $1$.\nTotal variation is additive over adjacent intervals: $V_a^c(f) = V_a^b(f) + V_b^c(f)$ for $a \u003c b \u003c c$.\n\nNon-example of finite total variation: the indicator function of the rationals on $[0, 1]$.\nA partition alternating between rational and irrational points with $n$ subintervals gives the sum $n$, so the total variation is $\\infty$.\n\nIf $f$ is continuously differentiable on $[a, b]$, then $V_a^b(f) = \\int_a^b \\abs{f'(x)}\\, dx$.\n",
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