{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $D \\subseteq \\R$, let $f \\colon D \\to \\R$ be a [[function|0XHDTE5YMZ]], and let $A \\subseteq D$ be nonempty.\nThe *oscillation of $f$ on $A$* is\n$$\\operatorname{osc}_f(A) = \\sup_{x \\in A} f(x) - \\inf_{x \\in A} f(x) = \\sup \\set{\\abs{f(x) - f(y)} : x, y \\in A},$$\na value in $[0, \\infty]$.\nFor $x_0 \\in D$, the *oscillation of $f$ at $x_0$* is\n$$\\operatorname{osc}_f(x_0) = \\inf_{\\varepsilon \u003e 0} \\operatorname{osc}_f\\big(D \\cap (x_0 - \\varepsilon, x_0 + \\varepsilon)\\big).$$\n\n## Remarks\n\nThe quantity $\\operatorname{osc}_f(D \\cap (x_0 - \\varepsilon, x_0 + \\varepsilon))$ decreases as $\\varepsilon$ decreases, so the infimum is also the limit as $\\varepsilon \\to 0^+$.\nThe function $f$ is [[continuous at|0EEE2CZB2A]] $x_0$ if and only if $\\operatorname{osc}_f(x_0) = 0$.\nFor each $c \u003e 0$, the set $\\set{x \\in D : \\operatorname{osc}_f(x) \\ge c}$ is closed in $D$.\nSo the set of discontinuities of $f$ is a countable union of closed subsets of $D$.\n\nExample: let $f(x) = x^2$ on $\\R$ and $x_0 = 1$.\nOn $(1 - \\varepsilon, 1 + \\varepsilon)$ with $\\varepsilon \u003c 1$, the values fill $((1 - \\varepsilon)^2, (1 + \\varepsilon)^2)$, a spread of $4\\varepsilon$.\nThis tends to $0$, so $\\operatorname{osc}_f(1) = 0$, as continuity predicts.\n\nNon-example of continuity: the [[Dirichlet function|0T4EJ8ZJM9]], equal to $1$ on $\\Q$ and $0$ elsewhere, takes both values on every interval.\nSo its oscillation is $1$ at every point, and it is continuous nowhere.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "defines": "5XW4N8KY3Y",
    "kind": "definition",
    "msc": [
      "26A15"
    ],
    "name": "Oscillation of a function",
    "sources": [
      {
        "locator": "Definitions",
        "url": "https://en.wikipedia.org/wiki/Oscillation_(mathematics)"
      }
    ],
    "tag": "YMGHNBDZCP",
    "uses": [
      "0XHDTE5YMZ",
      "0BTMQ04556",
      "04KFTA34C5",
      "0GSZ7B0CGR",
      "0EEE2CZB2A",
      "0T4EJ8ZJM9",
      "0565EGK6XP"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/YMGHNBDZCP/",
  "links": [
    {
      "field": "defines",
      "target": "5XW4N8KY3Y"
    },
    {
      "field": "uses",
      "target": "0XHDTE5YMZ"
    },
    {
      "field": "uses",
      "target": "0BTMQ04556"
    },
    {
      "field": "uses",
      "target": "04KFTA34C5"
    },
    {
      "field": "uses",
      "target": "0GSZ7B0CGR"
    },
    {
      "field": "uses",
      "target": "0EEE2CZB2A"
    },
    {
      "field": "uses",
      "target": "0T4EJ8ZJM9"
    },
    {
      "field": "uses",
      "target": "0565EGK6XP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/YMGHNBDZCP.md",
  "msc": [
    "26A15"
  ],
  "name": "Oscillation of a function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/YMG/YMGHNBDZCP.md",
  "status": null,
  "tag": "YMGHNBDZCP",
  "uri": "m2://YMGHNBDZCP",
  "url": "https://math2.tamnd.com/wiki/oscillation-of-a-function-definition/",
  "writing": {
    "tag": "YMGHNBDZCP",
    "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
      }
    ]
  }
}