Barbălat’s lemma

Lemma (Barbălat).

Let f:(0,)R be Riemann integrablePlanetmathPlanetmath and uniformly continuousPlanetmathPlanetmath then


Note that if f is non-negative, then Riemann integrability is the same as being L1 in the sense of Lebesgue, but if f oscillates then the Lebesgue integralMathworldPlanetmath may not exist.

Further note that the uniform continuity is required to prevent sharp “spikes” that might prevent the limit from existing. For example suppose we add a spike of height 1 and area 2-n at every integer. Then the function is continuousMathworldPlanetmath and L1 (and thus Riemann integrable), but f(t) would not have a limit at infinity.


  • 1 Hartmut Logemann, Eugene P. Ryan. . The American Mathematical Monthly, 111(10):864–889, 2004.
Title Barbălat’s lemma
Canonical name BarbualatsLemma
Date of creation 2013-03-22 14:52:31
Last modified on 2013-03-22 14:52:31
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 7
Author jirka (4157)
Entry type Theorem
Classification msc 26A06
Synonym Barbalat’s lemma