{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "00KAFBR32E"
    },
    {
      "field": "uses",
      "target": "011HV2GAWF"
    },
    {
      "field": "uses",
      "target": "06SKCHDQT5"
    },
    {
      "field": "uses",
      "target": "0A8JXHYTJC"
    },
    {
      "field": "uses",
      "target": "0ATVXC3NCC"
    },
    {
      "field": "uses",
      "target": "0C5TP6RYSV"
    },
    {
      "field": "uses",
      "target": "0GR2APWTAC"
    },
    {
      "field": "uses",
      "target": "0NRQPQX3VA"
    },
    {
      "field": "uses",
      "target": "0PAQM3H57X"
    },
    {
      "field": "uses",
      "target": "0VGMHCF4SC"
    },
    {
      "field": "uses",
      "target": "0W0TZDRF9C"
    },
    {
      "field": "uses",
      "target": "0WS1A8E2P9"
    },
    {
      "field": "uses",
      "target": "1S20A3R659"
    },
    {
      "field": "uses",
      "target": "2YYCNFPX1F"
    },
    {
      "field": "uses",
      "target": "5KKCSZJC29"
    },
    {
      "field": "uses",
      "target": "6ZMQ1XZ44B"
    },
    {
      "field": "uses",
      "target": "7BVX2KF3QC"
    },
    {
      "field": "uses",
      "target": "8AJEMCZ29B"
    },
    {
      "field": "uses",
      "target": "8X7ZESGYBW"
    },
    {
      "field": "uses",
      "target": "91YF82M8AA"
    },
    {
      "field": "uses",
      "target": "A58Z1DQ593"
    },
    {
      "field": "uses",
      "target": "A7549DJW1N"
    },
    {
      "field": "uses",
      "target": "AEX1QP3178"
    },
    {
      "field": "uses",
      "target": "CRJSW18XVM"
    },
    {
      "field": "uses",
      "target": "D115PG3T7K"
    },
    {
      "field": "uses",
      "target": "FAHERE12SH"
    },
    {
      "field": "uses",
      "target": "H4FS63VY0Y"
    },
    {
      "field": "uses",
      "target": "KFW9841GQK"
    },
    {
      "field": "uses",
      "target": "MAM9H3SY3Y"
    },
    {
      "field": "uses",
      "target": "N35QM9G83G"
    },
    {
      "field": "uses",
      "target": "NPBYYK63FE"
    },
    {
      "field": "uses",
      "target": "TPJZCSXSVN"
    },
    {
      "field": "uses",
      "target": "XVQN0GT988"
    },
    {
      "field": "uses",
      "target": "Y5V5CN8FGN"
    },
    {
      "field": "uses",
      "target": "ZMEYSAHNX3"
    },
    {
      "field": "instance_of",
      "target": "BQXRD4GVKZ"
    },
    {
      "field": "specializes",
      "target": "T19P7J53A1"
    }
  ],
  "body": "\n## Definition\n\nLet $D \\subseteq \\mathbb{R}$ and let $f \\colon D \\to \\mathbb{R}$ be a function.\nThe function $f$ is continuous if it is [[continuous at|0EEE2CZB2A]] each point of $D$.\nFor a subset $S \\subseteq D$, $f$ is continuous on $S$ if its restriction to $S$ is continuous.\n\n## Remarks\n\nSums, [[products|0VGMHCF4SC]], [[quotients|0GR2APWTAC]], and [[compositions|0NRQPQX3VA]] of continuous functions are continuous.\n[[Polynomial functions are continuous|0C5TP6RYSV]]. So are the exponential function, the sine, and the cosine.\nOn a closed bounded interval, a continuous function has the [[intermediate value property|00KAFBR32E]] and [[attains its extreme values|0WS1A8E2P9]].\nMathlib defines it for topological spaces, as `Continuous`.\n\n## Explanations\n\n### L1 human\n\nA continuous function has no jumps or breaks at any point of its domain.\nOn an interval, we can draw its graph without lifting the pen.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "0X3WGX95FE",
    "inventory": [
      "56KCEN37MP"
    ],
    "kind": "definition",
    "msc": [
      "26A15"
    ],
    "name": "Continuous function",
    "sources": [
      {
        "locator": "Definition",
        "url": "https://en.wikipedia.org/wiki/Continuous_function"
      }
    ],
    "tag": "0P27W9RW0G",
    "uses": [
      "0EEE2CZB2A"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/0P27W9RW0G/",
  "links": [
    {
      "field": "defines",
      "target": "0X3WGX95FE"
    },
    {
      "field": "uses",
      "target": "0EEE2CZB2A"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/0P27W9RW0G.md",
  "msc": [
    "26A15"
  ],
  "name": "Continuous function",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0P2/0P27W9RW0G.md",
  "status": null,
  "tag": "0P27W9RW0G",
  "uri": "m2://0P27W9RW0G",
  "url": "https://tamnd.github.io/math2/wiki/continuous-function-definition/",
  "writing": {
    "tag": "0P27W9RW0G",
    "kind": "definition",
    "level": "W1",
    "next": [
      {
        "what": "edge not_instance_of in",
        "note": "one non-example that fails one clause",
        "have": 0,
        "want": 1
      }
    ]
  }
}