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  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}$ and let $f \\colon D \\to \\mathbb{R}$ be a function.\nThe function $f$ is uniformly continuous if the following holds.\nFor each $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ with $|f(x) - f(y)| \u003c \\varepsilon$ for all $x, y \\in D$ with $|x - y| \u003c \\delta$.\n\n## Remarks\n\nEach uniformly continuous function is continuous. In the [[epsilon-delta form|0W8YHHGCJG]] of continuity, $\\delta$ may depend on the point. Here one $\\delta$ works for all points.\nThe function $x^2$ is continuous on $\\mathbb{R}$ but [[not uniformly continuous|0A8JXHYTJC]].\nMathlib defines it for uniform spaces, as `UniformContinuous`.\nEduard Heine published the first definition of uniform continuity, in 1870.\n\n## Explanations\n\n### L1 human\n\nFor a uniformly continuous function, one input tolerance works everywhere in the domain.\nThe graph has no places where it becomes steeper and steeper without bound.\n",
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