Darboux integral

Builds on Upper and lower Darboux sums Upper and lower Darboux integrals Partition of an interval

Definition

Let $a < b$ be real numbers and let $f \colon [a, b] \to \R$ be a bounded function. For a partition $P$ given by $a = x_0 < x_1 < \cdots < x_n = b$ , 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 $U(f, P) = \sum_{i=1}^{n} M_i (x_i - x_{i-1}), \qquad L(f, P) = \sum_{i=1}^{n} m_i (x_i - x_{i-1}).$ The upper and lower Darboux integrals are $\overline{I}(f) = \inf_P U(f, P)$ and $\underline{I}(f) = \sup_P L(f, P)$ . The function $f$ is Darboux integrable on $[a, b]$ if $\overline{I}(f) = \underline{I}(f)$ . Its Darboux integral $\int_a^b f(x)\, dx$ is then this common value.

Remarks

Example: $f(x) = x$ on $[0, 1]$ . For 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 $\lim_n U(f, P_n) = \lim_n \frac{n+1}{2n} = \frac12$ .

Non-example: the Dirichlet function, equal to $1$ on rationals and $0$ on irrationals in $[0, 1]$ . Every 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.

Riemann's criterion: $f$ is Darboux integrable if and only if for every $\varepsilon > 0$ there is a partition $P$ with $U(f, P) - L(f, P) < \varepsilon$ . A bounded function is Darboux integrable exactly when it is Riemann integrable, with the same integral.

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Partition of an interval definition
Sequence definition
Real sequence definition
Bounded sequence definition
Upper and lower Darboux sums definition
Upper and lower Darboux integrals definition
Darboux integral definition

Tag	`4XYXDYXQZR`
URI	`m2://4XYXDYXQZR`
Kind	definition
Writing	W0 stub
MSC	26A42
Defines	Darboux integral
Uses	Upper and lower Darboux sums, Upper and lower Darboux integrals, Partition of an interval
Changed	2 November 2026

Darboux integral

Definition

Remarks

Reading path

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