{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "017DH8WXTE"
    },
    {
      "field": "uses",
      "target": "03Z4JC4YRJ"
    }
  ],
  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be bounded.\nThen $f$ is [[Riemann integrable|0N1JQXZXHC]] if and only if its upper and lower [[Darboux integrals|09FQT057T7]] are equal.\nIn this case the common value is the [[Riemann integral|0QM4TXT3DR]] $\\int_a^b f(x)\\, dx$.\n\n## Remarks\n\nEach Riemann sum lies between the lower and upper Darboux sums. This gives one direction.\nFor the other direction, one shows that Darboux sums for partitions of small mesh are close to the Darboux integrals.\nSo the Riemann and the Darboux approaches define the same integral.\n\n## Explanations\n\n### L1 human\n\nThere are two standard ways to define the integral, with Riemann sums or with upper and lower sums.\nThey give the same integrable functions and the same values.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "26A42"
    ],
    "name": "Darboux and Riemann integrals agree",
    "role": "theorem",
    "sources": [
      {
        "locator": "Refinement of a partition and relation to Riemann integration",
        "url": "https://en.wikipedia.org/wiki/Darboux_integral"
      }
    ],
    "tag": "021XJAQVDC",
    "uses": [
      "09FQT057T7",
      "0N1JQXZXHC",
      "0QM4TXT3DR"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/021XJAQVDC/",
  "links": [
    {
      "field": "uses",
      "target": "09FQT057T7"
    },
    {
      "field": "uses",
      "target": "0N1JQXZXHC"
    },
    {
      "field": "uses",
      "target": "0QM4TXT3DR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/021XJAQVDC.md",
  "msc": [
    "26A42"
  ],
  "name": "Darboux and Riemann integrals agree",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/021/021XJAQVDC.md",
  "status": null,
  "tag": "021XJAQVDC",
  "uri": "m2://021XJAQVDC",
  "url": "https://math2.tamnd.com/wiki/darboux-and-riemann-integrals-agree/",
  "writing": {
    "tag": "021XJAQVDC",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}