{
  "aliases": [
    "bolzano-theorem"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0044CPHJBV"
    },
    {
      "field": "uses",
      "target": "06ZARTQTQE"
    },
    {
      "field": "uses",
      "target": "07BY47HG5S"
    },
    {
      "field": "uses",
      "target": "0C1VYVNV2N"
    },
    {
      "field": "uses",
      "target": "0D6FCW80A0"
    },
    {
      "field": "uses",
      "target": "0J29GM30Z7"
    },
    {
      "field": "uses",
      "target": "0N91N98EG5"
    },
    {
      "field": "uses",
      "target": "0V7E9KP6Q2"
    }
  ],
  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be [[continuous|0P27W9RW0G]].\nLet $y$ be a real number between $f(a)$ and $f(b)$.\nThen there is $c \\in [a, b]$ with $f(c) = y$.\n\n## Remarks\n\nAssume $f(a) \u003c y \u003c f(b)$. Let $c$ be the supremum of $\\{ x \\in [a, b] : f(x) \u003c y \\}$. Continuity gives $f(c) \\le y$ and $f(c) \\ge y$.\nThe proof needs the completeness of $\\mathbb{R}$. Over $\\mathbb{Q}$, the function $x^2 - 2$ changes sign on $[1, 2]$ but has no rational zero.\nIt gives the [[existence of $n$-th roots|0J29GM30Z7]], and it shows that each odd degree real polynomial has a real root.\nMathlib states it as `intermediate_value_Icc`.\n\n## History\n\nBernard Bolzano first proved the theorem in 1817.\n\n## Explanations\n\n### L1 human\n\nA continuous function on an interval cannot skip values.\nTo get from $f(a)$ to $f(b)$, its graph must cross every horizontal line in between.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "bolzano-theorem"
    ],
    "kind": "statement",
    "msc": [
      "26A15"
    ],
    "name": "Intermediate value theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q245098",
      "wikipedia": "Intermediate value theorem"
    },
    "sources": [
      {
        "locator": "Theorem",
        "url": "https://en.wikipedia.org/wiki/Intermediate_value_theorem"
      }
    ],
    "tag": "00KAFBR32E",
    "uses": [
      "0P27W9RW0G"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/00KAFBR32E/",
  "links": [
    {
      "field": "uses",
      "target": "0P27W9RW0G"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/00KAFBR32E.md",
  "msc": [
    "26A15"
  ],
  "name": "Intermediate value theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/00K/00KAFBR32E.md",
  "status": null,
  "tag": "00KAFBR32E",
  "uri": "m2://00KAFBR32E",
  "url": "https://math2.tamnd.com/wiki/intermediate-value-theorem/",
  "writing": {
    "tag": "00KAFBR32E",
    "kind": "statement",
    "level": "W1",
    "next": [
      {
        "what": "edge about out",
        "note": "the concepts that the statement is about",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge proves in",
        "note": "at least one proof node",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge instance_of or illustrates in",
        "note": "one example that shows the statement at work",
        "have": 0,
        "want": 1
      },
      {
        "what": "explanation L2",
        "have": 0,
        "want": 1
      }
    ]
  }
}