{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $I$ be an open interval and let $f \\colon I \\to \\mathbb{R}$ be convex.\nThen $f$ is [[continuous|0P27W9RW0G]] on $I$.\n\n## Remarks\n\nFix $y \\in I$ and points $x \u003c y \u003c z$ in $I$. By the [[slope inequalities|0BY5FTHEEE]], the slopes of chords from $y$ are bounded for points near $y$. So $|f(w) - f(y)| \\le C |w - y|$ near $y$.\nOn a closed interval this fails. The function on $[0, 1]$ with $f(0) = 1$ and $f(x) = 0$ for $x \u003e 0$ is convex but not continuous at $0$.\nA convex function on an open interval has one-sided derivatives at every point. Mathlib states continuity as `ConvexOn.continuousOn`, for open sets.\n\n## Explanations\n\n### L1 human\n\nA convex function cannot jump inside an open interval.\nA jump would break the condition that the chords lie above the graph.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "26A51"
    ],
    "name": "A convex function on an open interval is continuous",
    "role": "theorem",
    "sources": [
      {
        "locator": "Functions of one variable",
        "url": "https://en.wikipedia.org/wiki/Convex_function"
      }
    ],
    "tag": "011HV2GAWF",
    "uses": [
      "0BY5FTHEEE",
      "0P27W9RW0G"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/011HV2GAWF/",
  "links": [
    {
      "field": "uses",
      "target": "0BY5FTHEEE"
    },
    {
      "field": "uses",
      "target": "0P27W9RW0G"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/011HV2GAWF.md",
  "msc": [
    "26A51"
  ],
  "name": "A convex function on an open interval is continuous",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/011/011HV2GAWF.md",
  "status": null,
  "tag": "011HV2GAWF",
  "uri": "m2://011HV2GAWF",
  "url": "https://math2.tamnd.com/wiki/a-convex-function-on-an-open-interval-is-continuous/",
  "writing": {
    "tag": "011HV2GAWF",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}