{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "06B8BNJCD6"
    },
    {
      "field": "uses",
      "target": "07DCKDXRBP"
    }
  ],
  "body": "\n## Statement\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $f \\colon X \\to Y$ be a function.\nThen the following two conditions are equivalent:\n\n1. For each real $\\varepsilon \u003e 0$ there is a real $\\delta \u003e 0$ such that $d_Y(f(x), f(y)) \u003c \\varepsilon$ for all $x, y \\in X$ with $d_X(x, y) \u003c \\delta$.\n2. For all sequences $(x_n)$ and $(y_n)$ in $X$ with $d_X(x_n, y_n) \\to 0$, also $d_Y(f(x_n), f(y_n)) \\to 0$.\n\nThe map $f$ is uniformly continuous when it satisfies these conditions.\n\n## Remarks\n\nCondition 1 is the epsilon-delta definition of uniform continuity. Here $\\delta$ depends only on $\\varepsilon$, not on the points.\nEach uniformly continuous map is continuous. The converse fails: $x \\mapsto x^2$ on $\\mathbb{R}$ is continuous but not uniformly continuous. Take $x_n = n + 1/n$ and $y_n = n$; then $x_n^2 - y_n^2 \u003e 2$.\nOn a compact space, each continuous map to a metric space is uniformly continuous: this is the [[Heine–Cantor theorem|06B8BNJCD6]].\nEach [[Lipschitz map|0TMZ21WKCN]] is uniformly continuous.\n\n## Explanations\n\n### L1 human\n\nFor a uniformly continuous map, one input tolerance $\\delta$ works in the whole space.\nThe map $x \\mapsto x^2$ gets steeper and steeper, so near large $x$ you need a smaller and smaller $\\delta$, and no single $\\delta$ works.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "kind": "statement",
    "msc": [
      "54E40"
    ],
    "name": "Epsilon-delta uniform continuity",
    "role": "proposition",
    "sources": [
      {
        "locator": "Definition of uniform continuity",
        "url": "https://en.wikipedia.org/wiki/Uniform_continuity"
      },
      {
        "locator": "Other characterizations",
        "url": "https://en.wikipedia.org/wiki/Uniform_continuity"
      }
    ],
    "tag": "0FHSDCQACY",
    "uses": [
      "063W5JMY85",
      "06JYPJ4WHH"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0FHSDCQACY/",
  "links": [
    {
      "field": "uses",
      "target": "063W5JMY85"
    },
    {
      "field": "uses",
      "target": "06JYPJ4WHH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0FHSDCQACY.md",
  "msc": [
    "54E40"
  ],
  "name": "Epsilon-delta uniform continuity",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0FH/0FHSDCQACY.md",
  "status": null,
  "tag": "0FHSDCQACY",
  "uri": "m2://0FHSDCQACY",
  "url": "https://math2.tamnd.com/wiki/epsilon-delta-uniform-continuity/",
  "writing": {
    "tag": "0FHSDCQACY",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}