Open Access
2003-2004 A note on the Denjoy-Bourbaki theorem.
Constantin P. Niculescu, Florin Popovici
Author Affiliations +
Real Anal. Exchange 29(2): 639-646 (2003-2004).


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.


Download Citation

Constantin P. Niculescu. Florin Popovici. "A note on the Denjoy-Bourbaki theorem.." Real Anal. Exchange 29 (2) 639 - 646, 2003-2004.


Published: 2003-2004
First available in Project Euclid: 7 June 2006

zbMATH: 1072.26004
MathSciNet: MR2083802

Primary: 26A24 , 26A39
Secondary: 26A46 , 26D10

Keywords: differentiable function , generalized absolute continuity , negligible variation

Rights: Copyright © 2003 Michigan State University Press

Vol.29 • No. 2 • 2003-2004
Back to Top