Real Analysis Exchange

A Generalization of the Riemann-Lebesgue Theorem for Riemann Integrability

Otgonbayar Uuye

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A classical theorem of Riemann and Lebesgue says that a bounded function defined on a compact interval is Riemann integrable if and only if it is continuous almost everywhere. In this note, we generalize their result and show that the difference between the upper and lower Riemann integrals of a not necessarily Riemann integrable function equals the upper Riemann integral of its oscillation function.

Article information

Real Anal. Exchange, Volume 45, Number 2 (2020), 481-486.

First available in Project Euclid: 30 June 2020

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Digital Object Identifier

Primary: 26A42: Integrals of Riemann, Stieltjes and Lebesgue type [See also 28-XX]
Secondary: 26A06: One-variable calculus

Riemann integrability Riemann-Lebesgue theorem Oscillation function


Uuye, Otgonbayar. A Generalization of the Riemann-Lebesgue Theorem for Riemann Integrability. Real Anal. Exchange 45 (2020), no. 2, 481--486. doi:10.14321/realanalexch.45.2.0481.

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