{
  "aliases": [
    "intermediate-value-theorem"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $X$ be a [[connected space|0D939CMME7]], let $f \\colon X \\to \\mathbb{R}$ be [[continuous|0SY9ECQ19Q]], and let $a, b \\in X$.\nThen for each real $c$ between $f(a)$ and $f(b)$ there is $x \\in X$ with $f(x) = c$.\n\n## Remarks\n\nProof: $f(X)$ is connected, by [[continuous image of a connected set is connected|0EPD87YVG8]]. Thus it is an interval, by [[connected subsets of the real line are intervals|0B9ZQFMQNV]], and it contains $c$.\nFor $X = [a, b]$, this is the intermediate value theorem of calculus: a continuous $f \\colon [a, b] \\to \\mathbb{R}$ takes each value between $f(a)$ and $f(b)$.\nFor example, each polynomial of odd degree with real coefficients has a real root.\n\n## Explanations\n\n### L1 human\n\nA continuous real function on a connected space cannot jump over a value. To get from $f(a)$ to $f(b)$, it passes through all values in between.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "intermediate-value-theorem"
    ],
    "kind": "statement",
    "msc": [
      "54D05"
    ],
    "name": "Intermediate value theorem for connected spaces",
    "role": "theorem",
    "sources": [
      {
        "locator": "General metric and topological spaces",
        "url": "https://en.wikipedia.org/wiki/Intermediate_value_theorem"
      },
      {
        "locator": "Theorem",
        "url": "https://en.wikipedia.org/wiki/Intermediate_value_theorem"
      }
    ],
    "tag": "098C0KZZAP",
    "uses": [
      "0EPD87YVG8",
      "0B9ZQFMQNV",
      "0SY9ECQ19Q"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/098C0KZZAP/",
  "links": [
    {
      "field": "uses",
      "target": "0EPD87YVG8"
    },
    {
      "field": "uses",
      "target": "0B9ZQFMQNV"
    },
    {
      "field": "uses",
      "target": "0SY9ECQ19Q"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/098C0KZZAP.md",
  "msc": [
    "54D05"
  ],
  "name": "Intermediate value theorem for connected spaces",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/098/098C0KZZAP.md",
  "status": null,
  "tag": "098C0KZZAP",
  "uri": "m2://098C0KZZAP",
  "url": "https://math2.tamnd.com/wiki/intermediate-value-theorem-for-connected-spaces/",
  "writing": {
    "tag": "098C0KZZAP",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}