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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|0N1JQXZXHC]] if and only if the set of points where $f$ is not [[continuous|0EEE2CZB2A]] has Lebesgue measure zero.\n\n## Remarks\n\nA set $E \\subseteq \\mathbb{R}$ has measure zero if for each $\\varepsilon \u003e 0$ it is covered by countably many open intervals of total length less than $\\varepsilon$.\nCountable sets have measure zero. So [[Thomae's function|0HJ28H9Y95]] is integrable, and so is each monotone function.\nThe [[Dirichlet function|03Z4JC4YRJ]] is discontinuous everywhere, so it is not integrable.\nGiuseppe Vitali and Henri Lebesgue proved the criterion independently, in 1907.\n\n## Explanations\n\n### L1 human\n\nA bounded function is Riemann integrable exactly when it is continuous at almost all points.\nThe bad points must form a set of total length zero.\n",
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  "name": "Lebesgue's criterion for Riemann integrability",
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