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Barb\u{a}lat's lemma
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(Theorem)
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Note that if $f$ is non-negative, then Riemann integrability is the same as being $L^1$ in the sense of Lebesgue, but if $f$ oscillates then the Lebesgue integral 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 continuous and $L^1$ (and thus Riemann
integrable), but $f(t)$ would not have a limit at infinity.
- 1
- Hartmut Logemann, Eugene P. Ryan. Asymptotic behaviour of nonlinear systems. The American Mathematical Monthly, 111(10):864-889, 2004.
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"Barb\u{a}lat's lemma" is owned by jirka.
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(view preamble | get metadata)
| Other names: |
Barbalat's lemma |
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Cross-references: infinity, continuous, function, integer, area, height, limit, uniform continuity, Lebesgue integral, Riemann, uniformly continuous, Riemann integrable
This is version 4 of Barb\u{a}lat's lemma, born on 2004-12-10, modified 2005-04-09.
Object id is 6552, canonical name is BarbualatsLemma.
Accessed 4583 times total.
Classification:
| AMS MSC: | 26A06 (Real functions :: Functions of one variable :: One-variable calculus) |
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Pending Errata and Addenda
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