{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nWe bound each difference quotient with the mean value theorem.\n\n1. Let $x, y \\in I$. We must show $|f(x) - f(y)| \\le K |x - y|$.\n2. If $x = y$, both sides are $0$. The inequality is symmetric in $x$ and $y$, so we may assume $x \u003c y$.\n3. Since $I$ is an [[interval|0TK8Y3J5RW]], every $z$ with $x \\le z \\le y$ lies in $I$. So $[x, y] \\subseteq I$.\n4. Each $z \\in (x, y)$ is an interior point of $I$, because the open interval $(x, y)$ contains $z$ and lies in $I$.\n5. So $f$ is differentiable on $(x, y)$, with $|f'(z)| \\le K$ there. Also $f$ is continuous on $[x, y]$.\n6. By the [[mean value theorem|0BARQY52N5]], there is $c \\in (x, y)$ with $f(y) - f(x) = f'(c)(y - x)$.\n7. Taking absolute values, $|f(y) - f(x)| = |f'(c)| \\, |y - x| \\le K |y - x|$, by step 5.\n8. This holds for all $x, y \\in I$, so $f$ is [[Lipschitz with constant $K$|0QNVQ4VD9T]].\n\n## Idea\n\nThe mean value theorem says each chord slope of $f$ equals a value of $f'$. If every value of $f'$ has size at most $K$, every chord slope does too. That is exactly the Lipschitz condition.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "26A16"
    ],
    "name": "Proof that a function with bounded derivative is Lipschitz",
    "proves": "0P64VE79X3",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Lipschitz_continuity"
      }
    ],
    "tag": "HD08YKADB1",
    "uses": [
      "0BARQY52N5",
      "0QNVQ4VD9T",
      "0TK8Y3J5RW"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/HD08YKADB1/",
  "links": [
    {
      "field": "proves",
      "target": "0P64VE79X3"
    },
    {
      "field": "uses",
      "target": "0BARQY52N5"
    },
    {
      "field": "uses",
      "target": "0QNVQ4VD9T"
    },
    {
      "field": "uses",
      "target": "0TK8Y3J5RW"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/HD08YKADB1.md",
  "msc": [
    "26A16"
  ],
  "name": "Proof that a function with bounded derivative is Lipschitz",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HD0/HD08YKADB1.md",
  "status": null,
  "tag": "HD08YKADB1",
  "uri": "m2://HD08YKADB1",
  "url": "https://math2.tamnd.com/wiki/proof-that-a-function-with-bounded-derivative-is-lipschitz/",
  "writing": {
    "tag": "HD08YKADB1",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}