Real Analysis Exchange

A note on the Denjoy-Bourbaki theorem.

Constantin P. Niculescu and Florin Popovici

Full-text: Open access


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

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

First available in Project Euclid: 7 June 2006

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Differentiable function generalized absolute continuity negligible variation


Niculescu, Constantin P.; Popovici, Florin. A note on the Denjoy-Bourbaki theorem. Real Anal. Exchange 29 (2003), no. 2, 639--646.

Export citation


  • R. G. Bartle, A Modern Theory of Integration, Graduate Studies in Mathematics, 32, Amer. Math. Soc., Providence, R. I., 2001.
  • N. Bourbaki, Fonctions d'une variable réelle (Théorie élémentaire), Hermann, Paris, 1949.
  • A. Denjoy, Mémoire sur les nombres derivés des fonctions continues, Journal de Mathématiques Pures et Appliquées, 7th Series, 1 (1915), 105–240.
  • J. Dieudonné, Foundations of Modern Analysis, Academic Press, New York and London, 1960.
  • R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Graduate Studies in Mathematics, 4, Amer. Math. Soc., Providence, R. I., 1994.
  • K. S. K. Iyengar, Note on an inequality, Math. Student, 6 (1938), 75–76.
  • J. Pečarić, F. Proschan and Y. L. Tong, Convex Functions, Partial Ordering and Statistical Applications, Academic Press, 1992.