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  "body": "\n## Definition\n\nLet $\\gamma \\colon [a, b] \\to \\mathbb{R}^n$ be a [[parametrised curve|0JF1XXGWB4]].\nFor a partition $P$ given by $a = t_0 \u003c t_1 \u003c \\dots \u003c t_k = b$, let\n$$L(\\gamma, P) = \\sum_{i = 1}^{k} \\|\\gamma(t_i) - \\gamma(t_{i-1})\\|$$\nbe the length of the inscribed polygonal path.\nThe curve $\\gamma$ is rectifiable if $\\sup_P L(\\gamma, P) \u003c \\infty$, where the supremum is over all partitions of $[a, b]$.\nThen the length of $\\gamma$ is $\\ell(\\gamma) = \\sup_P L(\\gamma, P)$.\n\n## Remarks\n\nIf $\\gamma$ is of class $C^1$, then it is rectifiable and $\\ell(\\gamma) = \\int_a^b \\|\\gamma'(t)\\| \\, dt$.\nA refinement of $P$ does not decrease $L(\\gamma, P)$, by the triangle inequality.\nThe graph of $f(x) = x \\sin(1/x)$ for $0 \u003c x \\le 1$, $f(0) = 0$, is a continuous curve that is not rectifiable. The Koch curve is another example.\nA curve is rectifiable if and only if each component $\\gamma_i$ has bounded variation on $[a, b]$.\n\n## Explanations\n\n### L1 human\n\nApproximate the curve by a chain of straight segments between points on it, and add up their lengths.\nFiner chains give longer totals. If the totals stay bounded, the curve has a finite length, and the length is their least upper bound.\n",
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