{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0W0TZDRF9C"
    }
  ],
  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}$, let $f \\colon D \\to \\mathbb{R}$ be a function, and let $S \\subseteq D$.\nThe function $f$ is uniformly continuous on $S$ if the following holds.\nFor each $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ with $|f(x) - f(y)| \u003c \\varepsilon$ for all $x, y \\in S$ with $|x - y| \u003c \\delta$.\n\n## Remarks\n\nSo $f$ is uniformly continuous on $S$ when its restriction to $S$ is [[uniformly continuous|0Q4DR37DVE]].\nA function can be uniformly continuous on a set but not on a larger set. For example, $x^2$ is uniformly continuous on $[0, 1]$ but not on $\\mathbb{R}$.\nMathlib defines it as `UniformContinuousOn`.\n\n## Explanations\n\n### L1 human\n\nUniform continuity on $S$ only looks at pairs of points inside $S$.\nPoints outside $S$ play no role.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "07RB7B04HX",
    "kind": "definition",
    "msc": [
      "26A15"
    ],
    "name": "Uniform continuity on a set",
    "sources": [
      {
        "locator": "Definition of uniform continuity",
        "url": "https://en.wikipedia.org/wiki/Uniform_continuity"
      }
    ],
    "tag": "0A3CK6DVVB",
    "uses": [
      "0Q4DR37DVE"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0A3CK6DVVB/",
  "links": [
    {
      "field": "defines",
      "target": "07RB7B04HX"
    },
    {
      "field": "uses",
      "target": "0Q4DR37DVE"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0A3CK6DVVB.md",
  "msc": [
    "26A15"
  ],
  "name": "Uniform continuity on a set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0A3/0A3CK6DVVB.md",
  "status": null,
  "tag": "0A3CK6DVVB",
  "uri": "m2://0A3CK6DVVB",
  "url": "https://math2.tamnd.com/wiki/uniform-continuity-on-a-set-definition/",
  "writing": {
    "tag": "0A3CK6DVVB",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}