 
Summary: Z .JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 213, 487 495 1997
ARTICLE NO. AY975552
Rolle's Theorem and Negligibility of Points in Infinite
Dimensional Banach Spaces*
D. Azagra,
J. Gomez,
and J. A. Jaramillo§
´
Departamento de Analisis Matematico, Facultad de Ciencias Matematicas,´ ´ ´
Uni¨ersidad Complutense, Madrid, 28040, Spain
Submitted by Richard M. Aron
Received July 17, 1996
In this note we prove that if a differentiable function oscillates between y and
on the boundary of the unit ball then there exists a point in the interior of the
ball in which the differential of the function has norm equal or less than . This
kind of approximate Rolle's theorem is interesting because an exact Rolle's
theorem does not hold in many infinite dimensional Banach spaces. A characteri
zation of those spaces in which Rolle's theorem does not hold is given within a
large class of Banach spaces. This question is closely related to the existence of C1
Ä 4diffeomorphisms between a Banach space X and X _ 0 which are the identity out
