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  "body": "\n## Definition\n\nLet $I \\subseteq \\R$ be an interval, let $0 \u003c \\alpha \\le 1$, and let $M \u003e 0$.\nThe class $\\mathrm{Lip}_M \\alpha$ on $I$ is the set of functions $f \\colon I \\to \\R$ such that\n$$\\abs{f(x) - f(y)} \\le M \\abs{x - y}^\\alpha \\quad \\text{for all } x, y \\in I.$$\nThe *Lipschitz class* $\\mathrm{Lip}\\,\\alpha$ on $I$ is the union of the classes $\\mathrm{Lip}_M \\alpha$ over all $M \u003e 0$.\n\n## Remarks\n\nExample: $f(x) = \\sqrt{x}$ on $[0, 1]$ lies in $\\mathrm{Lip}_1 \\tfrac12$.\nFor $0 \\le y \\le x$ one has $\\sqrt{x} - \\sqrt{y} \\le \\sqrt{x - y}$, since squaring gives $x + y - 2\\sqrt{xy} \\le x - y$, that is $y \\le \\sqrt{xy}$.\n\nNon-example: the same $f$ is not in $\\mathrm{Lip}\\,1$, because $\\abs{f(x) - f(0)} / \\abs{x - 0} = x^{-1/2}$ is unbounded as $x \\to 0$.\n\nIn terms of the [[modulus of continuity|AXCJBSQZWF]] $\\omega(f, \\delta)$, the condition $f \\in \\mathrm{Lip}_M \\alpha$ says $\\omega(f, \\delta) \\le M \\delta^\\alpha$ for all $\\delta \u003e 0$.\n$\\mathrm{Lip}\\,1$ is the class of [[Lipschitz continuous|0QNVQ4VD9T]] functions, and $\\mathrm{Lip}\\,\\alpha$ for $\\alpha \u003c 1$ is the class of [[Hölder continuous|012PBA0FMD]] functions of exponent $\\alpha$.\nThe restriction $\\alpha \\le 1$ is natural: for $\\alpha \u003e 1$ the condition forces $f' = 0$, so only constant functions remain.\n",
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