{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "065XW2V6C3"
    },
    {
      "field": "uses",
      "target": "0B89D93YA1"
    },
    {
      "field": "uses",
      "target": "0M0RZ3HSB0"
    },
    {
      "field": "uses",
      "target": "0RDP2HEFEW"
    },
    {
      "field": "uses",
      "target": "0RKW2X63YR"
    }
  ],
  "body": "\n## Statement\n\nLet $I$ be an interval and let $f \\colon I \\to \\mathbb{R}$ be continuous and [[strictly increasing|0TR9SR9FTB]].\nThen $J = f(I)$ is an interval, and the inverse function $f^{-1} \\colon J \\to I$ is continuous and strictly increasing.\n\n## Remarks\n\nThe set $J$ is an interval by the [[intermediate value theorem|00KAFBR32E]].\nThe inverse is strictly increasing and its image is the interval $I$. A monotone function whose image is an interval has no jumps, so $f^{-1}$ is continuous.\nThe same holds for strictly decreasing functions.\nThis gives the continuity of $n$-th roots, of the [[logarithm|0044CPHJBV]], and of the inverse trigonometric functions.\n\n## Explanations\n\n### L1 human\n\nThe graph of $f^{-1}$ is the graph of $f$ reflected in the line $y = x$.\nA rising unbroken graph stays rising and unbroken after the reflection.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "26A48"
    ],
    "name": "Continuity of the inverse of a strictly monotone continuous function",
    "role": "theorem",
    "sources": [
      {
        "locator": "Standard inverse functions",
        "url": "https://en.wikipedia.org/wiki/Inverse_function"
      }
    ],
    "tag": "07BY47HG5S",
    "uses": [
      "0TR9SR9FTB",
      "00KAFBR32E",
      "0KJXHN0TW0"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/07BY47HG5S/",
  "links": [
    {
      "field": "uses",
      "target": "0TR9SR9FTB"
    },
    {
      "field": "uses",
      "target": "00KAFBR32E"
    },
    {
      "field": "uses",
      "target": "0KJXHN0TW0"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/07BY47HG5S.md",
  "msc": [
    "26A48"
  ],
  "name": "Continuity of the inverse of a strictly monotone continuous function",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/07B/07BY47HG5S.md",
  "status": null,
  "tag": "07BY47HG5S",
  "uri": "m2://07BY47HG5S",
  "url": "https://math2.tamnd.com/wiki/continuity-of-the-inverse-of-a-strictly-monotone-continuous-function/",
  "writing": {
    "tag": "07BY47HG5S",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}