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  "body": "\n## Summary\n\nA modulus of continuity of a function $f$ is a function $\\omega$ that bounds how much $f$ can change: $\\abs{f(x) - f(y)} \\le \\omega(\\abs{x - y})$.\nThe best such bound is the function $\\delta \\mapsto \\sup \\set{\\abs{f(x) - f(y)} : \\abs{x - y} \\le \\delta}$.\nA function is [[uniformly continuous|0Q4DR37DVE]] exactly when this bound tends to $0$ with $\\delta$.\nModuli of continuity measure the quality of continuity, and [[Lipschitz|0QNVQ4VD9T]] and [[Hölder|0WZTM0EFR6]] conditions are moduli of special forms.\n\n## Explanations\n\n### L1 human\n\nContinuity says that small changes of the input give small changes of the output.\nA modulus of continuity says how small, in numbers.\nIf two inputs are at distance $\\delta$, the outputs differ by at most $\\omega(\\delta)$.\nFor $f(x) = 3x$ one can take $\\omega(\\delta) = 3\\delta$, and for $f(x) = \\sqrt x$ on $[0, \\infty)$ one can take $\\omega(\\delta) = \\sqrt \\delta$.\n",
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