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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.\nFor a [[partition|069P20KYQT]] $P$ given by $a = x_0 \u003c x_1 \u003c \\cdots \u003c x_n = b$, put\n$$V(f, P) = \\sum_{i=1}^{n} \\abs{f(x_i) - f(x_{i-1})}.$$\nThe [[total variation|8QA5DEWNYD]] of $f$ on $[a, b]$ is $V_a^b(f) = \\sup_P V(f, P)$, where the supremum runs over all partitions of $[a, b]$.\nThe function $f$ has *bounded variation* on $[a, b]$ if $V_a^b(f) \u003c \\infty$.\n\n## Remarks\n\nExample: a nondecreasing function $f$ on $[a, b]$ has bounded variation.\nFor every partition the sum telescopes: $V(f, P) = \\sum_i (f(x_i) - f(x_{i-1})) = f(b) - f(a)$, so $V_a^b(f) = f(b) - f(a)$.\n\nNon-example: let $f(0) = 0$ and $f(x) = x \\sin(1/x)$ for $0 \u003c x \\le 1$.\nThe function $f$ is continuous, but it does not have bounded variation.\nAt the points $x_k = 1/((k + \\frac12)\\pi)$ the values are $\\pm x_k$ with alternating signs, so a partition through $x_1, \\dots, x_N$ gives $V(f, P) \\ge \\sum_{k=1}^{N} x_k$, and this sum diverges like the harmonic series.\n\nThe set of functions of bounded variation on $[a, b]$ is often written $BV([a, b])$.\nEvery function of bounded variation is bounded, since $\\abs{f(x)} \\le \\abs{f(a)} + V_a^b(f)$.\n",
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