Riemann integrable definition of Riemann integrable in the Free Online Encyclopedia.

Riemann integral [′rē‚män ‚int·ə·grəl]

(mathematics)

The Riemann integral of a real function ƒ(x) on an interval (a,b) is the unique limit (when it exists) of the sum of ƒ(a_i)(x_i-x_i-1),i= 1, …,n, taken over all partitions of (a,b),a=x₀<a₁<x₁< ⋯ <a_n<x_n=b, as the maximum distance betweenx_iandx_i-1tends to zero.

Riemann Integral

the usual definite integral. The definition of the Riemann integral was in essence given by A. Cauchy in 1823. Cauchy, however, applied his definition to continuous functions. In a paper written in 1853 and published in 1867, B. Riemann was the first to point out the necessary and sufficient conditions for the existence of the definite integral. In modern terms, these conditions can be stated as follows: for the definite integral of a function to exist over some interval, it is necessary and sufficient that (1) the interval be finite, (2) the function to be bounded on the interval, and (3) the set of discontinuities of the function on the interval have Lebesgue measure zero (seeMEASURE OF A SET).

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No references found	2) provided u is L-Lipschitzian and f is Riemann integrable and with the property that there exists the constants m, M [member of] R such that m [less than or equal to] f (x) [less than or equal to] M for any x [member of] [a, b]. A generalisation of Cerone's identity and applications by Dragomir, S.S. / Tamsui Oxford Journal of Mathematical Sciences	riebeckite asbestos riebungsbreccia Riecke's principle Riedel's disease riedenite Riefenstahl Riefenstahl, Leni Riefler clock Riegel Rieger, František Ladislav Rieger, Frantisek Ladislav Riegger, Wallingford Riegl, Alois Riegler's test Riego y Nunez, Rafael Riego y Núñez, Rafael Riego y Nuñez, Rafael del Riehen Riehl, Alois Rieke diagram Riel's rebellions Riel, Louis Riemann Riemann function Riemann hypothesis Riemann integrable Riemann Integral Riemann mapping theorem Riemann method Riemann P function Riemann space Riemann Sphere Riemann Surface Riemann surfaces Riemann tensors Riemann zeta function Riemann, Bernhard Riemann, Georg Friedrich Bernhard Riemann, Hugo Riemann-Christoffel tensor Riemann-Lebesgue lemma Riemann-Stieltjes integral Riemannian curvature Riemannian Geometry Riemannian manifold Riemannian Space Riemannian space-time Riemenschneider, Tilman Rienacker, Gunter Rienäcker, Günter Rienzi	Rieke diagram riel riel riel riel Riel Rebellion Riel's rebellions Riel, Louis Riel, Louis Riel, Louis RIELA Rieles Rieles riels riels Riem Riem RIEMA Rieman sphere Riemann Riemann Riemann (Georg Friedrich) Bernhard Riemann conjecture Riemann curvature tensor Riemann for Anti-Dummies Riemann function Riemann geometry Riemann geometry Riemann hypotheses Riemann hypothesis Riemann integrable Riemann integral Riemann integration Riemann lemma Riemann mapping theorem Riemann method Riemann multiple integral Riemann P function Riemann space Riemann space Riemann sphere Riemann surface Riemann surfaces Riemann tensors Riemann theta function Riemann theta function Riemann zeta Riemann zeta function Riemann zeta function zeros Riemann zeta hypothesis Riemann zeta-function Riemann zeta-hypothesis Riemann's hypothesis Riemann's mapping theorem Riemann's surface Riemann's zeta function Riemann, (Georg Friedrich) Bernhard Riemann, (Georg Friedrich) Bernhard

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2) provided u is L-Lipschitzian and f is Riemann integrable and with the property that there exists the constants m, M [member of] R such that m [less than or equal to] f (x) [less than or equal to] M for any x [member of] [a, b].

A generalisation of Cerone's identity and applications by Dragomir, S.S. / Tamsui Oxford Journal of Mathematical Sciences

Rieger, František Ladislav

Rieger, Frantisek Ladislav