A note on the Denjoy-Bourbaki theorem.

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.