Real Analysis Exchange

A note on the Denjoy-Bourbaki theorem.

Constantin P. Niculescu and Florin Popovici

Full-text: Open access

Abstract

WWe prove the following extension of the Mean Value Theorem. Let E be a Banach space and let $F:[a,b]\rightarrow E$and $\varphi :[a,b]\rightarrow\mathbb{R}$ be two functions for which there exists a subset $A\subset\lbrack a,b]$ such that:

i) $F$ and $\varphi$ have negligible variation on $A$,

ii) $F$ and $\varphi$ are differentiable on $[a,b]\setminus A$ and $\left\Vert F^{\prime}\right\Vert \leq\varphi^{\prime}$ on $[a,b]\setminus A$.

Then $\left\Vert F(b)-F(a)\right\Vert \leq\varphi(b)-\varphi(a).$

Several applications are included.

Article information

Source
Real Anal. Exchange, Volume 29, Number 2 (2003), 639-646.

Dates
First available in Project Euclid: 7 June 2006

Permanent link to this document
https://projecteuclid.org/euclid.rae/1149698554

Mathematical Reviews number (MathSciNet)
MR2083802

Zentralblatt MATH identifier
1072.26004

Subjects
Primary: 26A24: Differentiation (functions of one variable): general theory, generalized derivatives, mean-value theorems [See also 28A15] 26A39: Denjoy and Perron integrals, other special integrals
Secondary: 26A46: Absolutely continuous functions 26D10: Inequalities involving derivatives and differential and integral operators

Keywords
Differentiable function generalized absolute continuity negligible variation

Citation

Niculescu, Constantin P.; Popovici, Florin. A note on the Denjoy-Bourbaki theorem. Real Anal. Exchange 29 (2003), no. 2, 639--646. https://projecteuclid.org/euclid.rae/1149698554


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References

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