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  "body": "\n## Definition\n\nLet $a \u003c b$ be real numbers and let $f \\colon [a, b] \\to \\R$ be a bounded function.\nFor a [[partition|069P20KYQT]] $P$ given by $a = x_0 \u003c x_1 \u003c \\cdots \u003c x_n = b$, let $M_i = \\sup_{[x_{i-1}, x_i]} f$ and $m_i = \\inf_{[x_{i-1}, x_i]} f$, and form the [[upper and lower Darboux sums|0QRQVH0CRW]]\n$$U(f, P) = \\sum_{i=1}^{n} M_i (x_i - x_{i-1}), \\qquad L(f, P) = \\sum_{i=1}^{n} m_i (x_i - x_{i-1}).$$\nThe [[upper and lower Darboux integrals|09FQT057T7]] are $\\overline{I}(f) = \\inf_P U(f, P)$ and $\\underline{I}(f) = \\sup_P L(f, P)$.\nThe function $f$ is *Darboux integrable* on $[a, b]$ if $\\overline{I}(f) = \\underline{I}(f)$.\nIts *Darboux integral* $\\int_a^b f(x)\\, dx$ is then this common value.\n\n## Remarks\n\nExample: $f(x) = x$ on $[0, 1]$.\nFor the partition $P_n$ into $n$ equal parts, $U(f, P_n) - L(f, P_n) = \\sum_{i=1}^{n} \\frac1n \\cdot \\frac1n = \\frac1n$, so the two Darboux integrals agree, and the integral is $\\lim_n U(f, P_n) = \\lim_n \\frac{n+1}{2n} = \\frac12$.\n\nNon-example: the [[Dirichlet function|03Z4JC4YRJ]], equal to $1$ on rationals and $0$ on irrationals in $[0, 1]$.\nEvery subinterval contains both kinds of numbers, so $U(f, P) = 1$ and $L(f, P) = 0$ for every $P$, and the Darboux integrals $1$ and $0$ differ.\n\nRiemann's criterion: $f$ is Darboux integrable if and only if for every $\\varepsilon \u003e 0$ there is a partition $P$ with $U(f, P) - L(f, P) \u003c \\varepsilon$.\nA bounded function is Darboux integrable exactly when it is [[Riemann integrable|0N1JQXZXHC]], with the same integral.\n",
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