{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "01XAY17QA3"
    },
    {
      "field": "uses",
      "target": "0X9XT05JVC"
    }
  ],
  "body": "\n## Definition\n\nLet $D$ be a set, let $f_n \\colon D \\to \\mathbb{R}$ for $n \\ge 1$ and $f \\colon D \\to \\mathbb{R}$ be functions, and let $S \\subseteq D$.\nThe sequence $(f_n)$ converges uniformly to $f$ on $S$ if for every $\\varepsilon \u003e 0$ there is $N$ with $|f_n(x) - f(x)| \u003c \\varepsilon$ for all $n \\ge N$ and all $x \\in S$.\n\n## Remarks\n\nThis is [[uniform convergence|08FPABDQ4C]] of the restrictions of $f_n$ and $f$ to $S$.\nA sequence can converge uniformly on each $[0, r]$ with $r \u003c 1$, but not on $[0, 1)$. The sequence $x^n$ is an example.\nMathlib defines it as `TendstoUniformlyOn`.\n\n## Explanations\n\n### L1 human\n\nWe often get uniform convergence only on part of the domain.\nThis definition measures the distance between $f_n$ and $f$ only on that part.\n",
  "commit": "545acf83a9b75c8ddc375ef8b23d2ae101a4d7e4",
  "front": {
    "defines": "0W1QMKJ9W8",
    "kind": "definition",
    "msc": [
      "40A30"
    ],
    "name": "Uniform convergence on a set",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Uniform_convergence"
      }
    ],
    "tag": "08HJRTG5N7",
    "uses": [
      "08FPABDQ4C"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/08HJRTG5N7/",
  "links": [
    {
      "field": "defines",
      "target": "0W1QMKJ9W8"
    },
    {
      "field": "uses",
      "target": "08FPABDQ4C"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/08HJRTG5N7.md",
  "msc": [
    "40A30"
  ],
  "name": "Uniform convergence on a set",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/08H/08HJRTG5N7.md",
  "status": null,
  "tag": "08HJRTG5N7",
  "uri": "m2://08HJRTG5N7",
  "url": "https://math2.tamnd.com/wiki/uniform-convergence-on-a-set-definition/",
  "writing": {
    "tag": "08HJRTG5N7",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}