{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f \\colon [a, b] \\to \\R$ be a function.\nThe function $f$ is *singular* if:\n\n1. $f$ is [[continuous|0P27W9RW0G]] on $[a, b]$;\n2. there is a set $N \\subseteq [a, b]$ of Lebesgue measure $0$ such that for every $x \\in [a, b] \\setminus N$ the [[derivative|0F1WR9C83K]] $f'(x)$ exists and $f'(x) = 0$;\n3. $f$ is not constant on $[a, b]$.\n\n## Remarks\n\nExample: the [[Cantor function|NPBYYK63FE]] $c$ on $[0, 1]$ is singular.\nIt is continuous, it is constant on each open interval removed in the construction of the Cantor set, so $c' = 0$ off the Cantor set, which has measure $0$, and $c(0) = 0 \\ne 1 = c(1)$.\n\nNon-example: the step function equal to $0$ on $[0, \\frac12)$ and $1$ on $[\\frac12, 1]$ satisfies clauses 2 and 3 but fails clause 1.\n\nConventions vary: some authors drop continuity, and some allow constants.\nA singular function is never [[absolutely continuous|ZT6F4C2YHK]], since an absolutely continuous function with derivative $0$ almost everywhere is constant.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "P5AWRD2793",
    "kind": "definition",
    "msc": [
      "26A30"
    ],
    "name": "Singular function",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Singular_function"
      }
    ],
    "tag": "2YYCNFPX1F",
    "uses": [
      "0P27W9RW0G",
      "0F1WR9C83K"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/2YYCNFPX1F/",
  "links": [
    {
      "field": "defines",
      "target": "P5AWRD2793"
    },
    {
      "field": "uses",
      "target": "0P27W9RW0G"
    },
    {
      "field": "uses",
      "target": "0F1WR9C83K"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/2YYCNFPX1F.md",
  "msc": [
    "26A30"
  ],
  "name": "Singular function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2YY/2YYCNFPX1F.md",
  "status": null,
  "tag": "2YYCNFPX1F",
  "uri": "m2://2YYCNFPX1F",
  "url": "https://math2.tamnd.com/wiki/singular-function-definition/",
  "writing": {
    "tag": "2YYCNFPX1F",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}