Azagra Rueda, DanielGómez Gil, JavierJaramillo Aguado, Jesús Ángel2023-06-202023-06-201997-09-150022-247X10.1006/jmaa.1997.5552https://hdl.handle.net/20.500.14352/57128In 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 characterization 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 diffeomorphisms between a Banach space X and X _ _04 which are the identity out of a ball, and we prove that such diffeomorphisms exist for every C1 smooth Banach space which can be linearly injected into a Banach space whose dual norm is locally uniformly rotund (LUR).engjournal articlehttp://www.sciencedirect.com/science/journal/0022247Xopen access517.98Rolle’s theorem in infinite-dimensional Banach spacesApproximate Rolle’s theoremContinuous norm whose dual norm is locally uniformly rotundC1 bump functionAnálisis funcional y teoría de operadores