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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be bounded.\nThen $f$ is Riemann integrable on $[a, b]$ if and only if the following holds.\nFor each $\\varepsilon \u003e 0$ there is a partition $P$ of $[a, b]$ with $U(f, P) - L(f, P) \u003c \\varepsilon$.\n\n## Remarks\n\nHere $U$ and $L$ are the upper and lower [[Darboux sums|0QRQVH0CRW]].\nThe condition says that the upper and lower Darboux integrals are equal. So it follows from the [[equality of Darboux and Riemann integrability|021XJAQVDC]].\nThis is the usual tool to prove that [[continuous|06SESV7GQY]] and [[monotone|04YKAYDQPW]] functions are integrable.\n\n## Explanations\n\n### L1 human\n\nA function is integrable when we can trap its graph between step functions whose areas differ by as little as we like.\n",
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