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  "body": "\n## Definition\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]], let $f \\colon X \\to Y$ be a function, and let $C \\ge 0$ and $\\alpha \u003e 0$ be real numbers.\nThe map $f$ is Hölder continuous with constant $C$ and exponent $\\alpha$ when\n$$d_Y\\bigl(f(x), f(y)\\bigr) \\le C \\, d_X(x, y)^{\\alpha} \\quad \\text{for all } x, y \\in X.$$\n\n## Remarks\n\nFor $\\alpha = 1$, this is the condition for a [[Lipschitz map|0TMZ21WKCN]] with constant $C$.\nThe map $x \\mapsto \\sqrt{x}$ on $[0, \\infty)$ is Hölder continuous with $C = 1$ and $\\alpha = 1/2$.\nA Hölder continuous map is uniformly continuous: for $\\varepsilon \u003e 0$ and $C \u003e 0$, take $\\delta = (\\varepsilon / C)^{1/\\alpha}$.\nFor a real function on an interval, Hölder continuity with an exponent $\\alpha \u003e 1$ forces the function to be constant.\n\n## Explanations\n\n### L1 human\n\nThe Hölder condition is a weaker form of the Lipschitz condition.\nIt allows a map to be steep, as $\\sqrt{x}$ is near $0$, but it still controls how fast the values can change.\n",
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