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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be bounded.\nThe upper Darboux integral of $f$ is the infimum of the upper [[Darboux sums|0QRQVH0CRW]] $U(f, P)$ over all partitions $P$ of $[a, b]$.\nThe lower Darboux integral of $f$ is the supremum of the lower Darboux sums $L(f, P)$ over all partitions $P$.\n\n## Remarks\n\nThe lower Darboux integral is at most the upper one.\nThe function is Darboux integrable if the two are equal. By [[a theorem|021XJAQVDC]], this is the same as [[Riemann integrability|0N1JQXZXHC]].\nFor the [[Dirichlet function|03Z4JC4YRJ]] on $[0, 1]$, the upper integral is $1$ and the lower integral is $0$.\n\n## Explanations\n\n### L1 human\n\nThe upper integral is the best upper estimate of the area by step functions above the graph.\nThe lower integral is the best lower estimate by step functions below the graph.\n",
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