{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "6HVAGKSQWB"
    },
    {
      "field": "uses",
      "target": "KV0GW9K7JS"
    },
    {
      "field": "uses",
      "target": "KZHNA19FSS"
    }
  ],
  "body": "\n## Summary\n\nA bump function is a [[smooth function|0V33BNKZ9S]] that vanishes outside a bounded region.\nSuch functions exist because some smooth functions are not analytic.\nThey are used to cut off, localize and glue functions.\nExamples are smooth partitions of unity, mollification, and test functions in the theory of distributions.\n\n## Explanations\n\n### L1 human\n\nPicture a smooth hill that rises from flat ground and returns to flat ground, with no corners anywhere.\nOutside the hill, the function is exactly zero.\nA polynomial cannot do this: a polynomial that is zero on an interval is zero everywhere.\nThe standard formula is $e^{-1/(1 - x^2)}$ for $\\abs{x} \u003c 1$ and $0$ elsewhere.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "26E10",
      "46F05"
    ],
    "name": "Bump function",
    "same_as": {
      "wikidata": "Q1143737",
      "wikipedia": "Bump function"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Bump_function"
      }
    ],
    "specializes": [
      "0V33BNKZ9S"
    ],
    "tag": "HS65DW0HXH"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/HS65DW0HXH/",
  "links": [
    {
      "field": "specializes",
      "target": "0V33BNKZ9S"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/HS65DW0HXH.md",
  "msc": [
    "26E10",
    "46F05"
  ],
  "name": "Bump function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HS6/HS65DW0HXH.md",
  "status": null,
  "tag": "HS65DW0HXH",
  "uri": "m2://HS65DW0HXH",
  "url": "https://math2.tamnd.com/wiki/bump-function/",
  "writing": {
    "tag": "HS65DW0HXH",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}