{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\nHere $f$ is increasing in the sense of a [[monotone function|0WFF6AZPBP]]: $x \\le y$ implies $f(x) \\le f(y)$.\n\n1. Let $S = \\set{f(x) : a \u003c x \u003c c}$. It is nonempty, since $\\frac{a + c}{2} \\in (a, c)$.\n2. For $a \u003c x \u003c c$ we have $f(x) \\le f(c)$, since $f$ is increasing. So $f(c)$ is an upper bound of $S$.\n3. By the [[completeness axiom|0AXWET80YW]], $S$ has a [[supremum|0BTMQ04556]] $L$. Since $f(c)$ is an upper bound, $L \\le f(c)$.\n4. The point $c$ is a [[limit point|07EGCNHAVW]] of $(a, b) \\cap (-\\infty, c) = (a, c)$. Indeed, for $\\delta \u003e 0$ the point $c - \\frac{1}{2}\\min(\\delta, c - a)$ lies in $(a, c)$ and within $\\delta$ of $c$.\n5. Let $\\varepsilon \u003e 0$. The number $L - \\varepsilon$ is not an upper bound of $S$. So there is $x_0 \\in (a, c)$ with $f(x_0) \u003e L - \\varepsilon$.\n6. Put $\\delta = c - x_0 \u003e 0$. Let $x \\in (a, b)$ with $c - \\delta \u003c x \u003c c$. Then $x_0 \u003c x \u003c c$.\n7. Since $f$ is increasing, $L - \\varepsilon \u003c f(x_0) \\le f(x)$. Since $x \\in (a, c)$, also $f(x) \\le L$. So $\\abs{f(x) - L} \u003c \\varepsilon$.\n8. By the definition of the [[one-sided limit|0909PWZR62]], steps 4 to 7 give $\\lim_{x \\to c^-} f(x) = L = \\sup_{a \u003c x \u003c c} f(x)$.\n9. Now let $T = \\set{f(x) : c \u003c x \u003c b}$. It is nonempty, and $f(c)$ is a lower bound of $T$, since $f$ is increasing.\n10. By the [[existence of the infimum|0EDH403M21]], $T$ has an [[infimum|04KFTA34C5]] $R$. Since $f(c)$ is a lower bound, $f(c) \\le R$.\n11. In the same way as step 4, $c$ is a limit point of $(c, b)$.\n12. Let $\\varepsilon \u003e 0$. The number $R + \\varepsilon$ is not a lower bound of $T$. So there is $x_1 \\in (c, b)$ with $f(x_1) \u003c R + \\varepsilon$.\n13. Put $\\delta = x_1 - c \u003e 0$. Let $x \\in (a, b)$ with $c \u003c x \u003c c + \\delta$. Then $c \u003c x \u003c x_1$, so $R \\le f(x) \\le f(x_1) \u003c R + \\varepsilon$. Thus $\\abs{f(x) - R} \u003c \\varepsilon$.\n14. So $\\lim_{x \\to c^+} f(x) = R = \\inf_{c \u003c x \u003c b} f(x)$.\n15. Steps 3, 8, 10 and 14 give the chain $\\lim_{x \\to c^-} f(x) = L \\le f(c) \\le R = \\lim_{x \\to c^+} f(x)$.\n\n## Idea\n\nTo the left of $c$, the values of $f$ rise but never pass $f(c)$, so they have a least upper bound. Once one value comes within $\\varepsilon$ of that bound, all later values up to $c$ stay within $\\varepsilon$, because $f$ is increasing. The right side is the mirror image with the infimum.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "26A48"
    ],
    "name": "Proof that one-sided limits of a monotone function exist",
    "proves": "0KJXHN0TW0",
    "sources": [
      {
        "locator": "Proposition 3.6.2",
        "url": "https://www.jirka.org/ra/html/sec_monotonefunc.html"
      }
    ],
    "tag": "PPP6BB77FP",
    "uses": [
      "0WFF6AZPBP",
      "0909PWZR62",
      "07EGCNHAVW",
      "0AXWET80YW",
      "0EDH403M21",
      "0BTMQ04556",
      "04KFTA34C5"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/PPP6BB77FP/",
  "links": [
    {
      "field": "proves",
      "target": "0KJXHN0TW0"
    },
    {
      "field": "uses",
      "target": "0WFF6AZPBP"
    },
    {
      "field": "uses",
      "target": "0909PWZR62"
    },
    {
      "field": "uses",
      "target": "07EGCNHAVW"
    },
    {
      "field": "uses",
      "target": "0AXWET80YW"
    },
    {
      "field": "uses",
      "target": "0EDH403M21"
    },
    {
      "field": "uses",
      "target": "0BTMQ04556"
    },
    {
      "field": "uses",
      "target": "04KFTA34C5"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/PPP6BB77FP.md",
  "msc": [
    "26A48"
  ],
  "name": "Proof that one-sided limits of a monotone function exist",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/PPP/PPP6BB77FP.md",
  "status": null,
  "tag": "PPP6BB77FP",
  "uri": "m2://PPP6BB77FP",
  "url": "https://math2.tamnd.com/wiki/proof-that-one-sided-limits-of-a-monotone-function-exist/",
  "writing": {
    "tag": "PPP6BB77FP",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}