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  "body": "\n## Definition\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be a function.\nThe function $f$ is Riemann integrable on $[a, b]$ if there is a real number $I$ with the following property.\nFor each $\\varepsilon \u003e 0$ there is $\\delta \u003e 0$ such that $|S(f, P, t) - I| \u003c \\varepsilon$ for each tagged partition $(P, t)$ of $[a, b]$ with [[mesh|04GM4NYXJB]] $\\| P \\| \u003c \\delta$.\nHere $S(f, P, t)$ is the [[Riemann sum|0HBY48G8G1]] of $f$ for $(P, t)$.\n\n## Remarks\n\nThe number $I$ is unique. It is the [[Riemann integral|0QM4TXT3DR]] of $f$.\nA Riemann integrable function is bounded.\n[[Continuous functions|06SESV7GQY]] and [[monotone functions|04YKAYDQPW]] on $[a, b]$ are Riemann integrable. The [[Dirichlet function|03Z4JC4YRJ]] is not.\n\n## Explanations\n\n### L1 human\n\nA function is Riemann integrable if its Riemann sums settle down to one number as the partitions get finer.\nIt does not matter which tags we choose in the pieces.\n",
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