{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0381S8Z5P0"
    }
  ],
  "body": "\n## Definition\n\nLet $(X, d_X)$ and $(Y, d_Y)$ be [[metric spaces|063W5JMY85]] and let $\\mathcal{F}$ be a set of functions from $X$ to $Y$.\nThe family $\\mathcal{F}$ is equicontinuous at a point $a \\in X$ when the property that follows holds.\nFor each real $\\varepsilon \u003e 0$ there is a real $\\delta \u003e 0$ such that $d_Y(f(x), f(a)) \u003c \\varepsilon$ for all $f \\in \\mathcal{F}$ and all $x \\in X$ with $d_X(x, a) \u003c \\delta$.\nThe family $\\mathcal{F}$ is equicontinuous when it is equicontinuous at each point of $X$.\n\n## Remarks\n\nThe number $\\delta$ depends on $\\varepsilon$ and $a$, but not on $f$.\nEach finite set of continuous functions is equicontinuous.\nThe family of all functions $f \\colon \\mathbb{R} \\to \\mathbb{R}$ that are [[Lipschitz|0TMZ21WKCN]] with constant $1$ is equicontinuous: take $\\delta = \\varepsilon$.\nThe family $f_n(x) = x^n$ on $[0, 1]$ is not equicontinuous at $x = 1$.\n\n## Explanations\n\n### L1 human\n\nEach continuous function has its own $\\delta$ for each $\\varepsilon$. An equicontinuous family has one $\\delta$ that works for all functions in the family.\nThe functions $x^n$ become steeper and steeper near $1$, so no common $\\delta$ exists there.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0XN66MZN0Y",
    "kind": "definition",
    "msc": [
      "54C35"
    ],
    "name": "Equicontinuity",
    "sources": [
      {
        "locator": "Equicontinuity between metric spaces",
        "url": "https://en.wikipedia.org/wiki/Equicontinuity"
      }
    ],
    "tag": "0MW2WXTHMC",
    "uses": [
      "063W5JMY85",
      "0EKGKY5TGH"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0MW2WXTHMC/",
  "links": [
    {
      "field": "defines",
      "target": "0XN66MZN0Y"
    },
    {
      "field": "uses",
      "target": "063W5JMY85"
    },
    {
      "field": "uses",
      "target": "0EKGKY5TGH"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0MW2WXTHMC.md",
  "msc": [
    "54C35"
  ],
  "name": "Equicontinuity",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0MW/0MW2WXTHMC.md",
  "status": null,
  "tag": "0MW2WXTHMC",
  "uri": "m2://0MW2WXTHMC",
  "url": "https://math2.tamnd.com/wiki/equicontinuity-definition/",
  "writing": {
    "tag": "0MW2WXTHMC",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "one example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}