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  "body": "\n## Definition\n\nLet $D \\subseteq \\R$ and let $f \\colon D \\to \\R$ be a [[function|0XHDTE5YMZ]].\nA *modulus of continuity* is a function $\\omega \\colon [0, \\infty) \\to [0, \\infty]$ that is nondecreasing, satisfies $\\omega(0) = 0$, and satisfies $\\omega(\\delta) \\to 0$ as $\\delta \\to 0^+$.\nThe function $\\omega$ is a modulus of continuity *for* $f$ if\n$$\\abs{f(x) - f(y)} \\le \\omega(\\abs{x - y}) \\quad \\text{for all } x, y \\in D.$$\nThe *optimal modulus of continuity* of $f$ is the function $\\omega_f \\colon [0, \\infty) \\to [0, \\infty]$ given by\n$$\\omega_f(\\delta) = \\sup \\set{\\abs{f(x) - f(y)} : x, y \\in D, \\ \\abs{x - y} \\le \\delta},$$\nwhere the [[supremum|0BTMQ04556]] is taken in the [[extended real line|0GSZ7B0CGR]].\n\n## Remarks\n\nThe function $f$ has a modulus of continuity if and only if $f$ is [[uniformly continuous|0Q4DR37DVE]], and then $\\omega_f$ is the smallest one.\nThe same definitions work for a map between metric spaces, with distances in place of $\\abs{x - y}$ and $\\abs{f(x) - f(y)}$.\n\nExample: let $f(x) = \\sqrt x$ on $D = [0, \\infty)$.\nFor $0 \\le y \\le x$, one has $\\sqrt x - \\sqrt y \\le \\sqrt{x - y}$, since squaring gives $x - 2\\sqrt{xy} + y \\le x - y$, that is $y \\le \\sqrt{xy}$.\nEquality holds for $y = 0$, so $\\omega_f(\\delta) = \\sqrt \\delta$, and this tends to $0$.\n\nNon-example: let $f(x) = x^2$ on $\\R$.\nFor $\\delta \u003e 0$, one has $f(x + \\delta) - f(x) = 2x\\delta + \\delta^2$, which is unbounded in $x$.\nSo $\\omega_f(\\delta) = \\infty$ for each $\\delta \u003e 0$, and $\\omega_f$ does not tend to $0$.\n\nThe modulus $\\omega(\\delta) = L\\delta$ gives the [[Lipschitz condition|0QNVQ4VD9T]] with constant $L$.\nThe modulus $\\omega(\\delta) = C\\delta^\\alpha$ with $0 \u003c \\alpha \\le 1$ gives the [[Hölder condition|0WZTM0EFR6]] of exponent $\\alpha$.\n",
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