Azagra Rueda, DanielGómez Gil, JavierJaramillo Aguado, Jesús Ángel2023-06-202023-06-201997-09-151. C. Bessaga, Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere, Bull. Acad. PoZon. Sci. Sér. Sci. Math. 14 (1966), 27-31. 2. C. Bessaga and A. Pe1czynski, Selected tapies in infinite-dimensional topology, in "Monografie Matematyczne," PWN, Warsaw, 1975. 3. R Deville, G. Godefroy, and V. Zizler, Smoothness and renormings in Banach spaces, in "Pitman Monographs and Surveys in Pure and Applied Mathematics," VoL 64, Longman, Harlow, 1993. 4. T. Dobrowolski, Smooth and R-analytic negligibility of subsets and extension of homeomorphism in Banach spaces, Studia Math. 65 (1979), 115-139. 5. 1. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N S.) 1, No. 3 (1979),443-474. 6. M. Fabian, P. Hájek, and J Vanderwerff, On smooth variational principies in Banach spaces, 1. Math. Anal. Appl. 197 (1996), 153-172. 7. J Bés and J Ferrera, private communication. 8. J Ferrer, Rolle's theorem fails in 12 , Amer. Math. Month1y 103, No. 2 (1996), 161-165. 9. R R Phelps, Convex functions, monotone operators and differentiability, in "Lecture Notes in Mathematics," VoL 1364, Springer-Verlag, BerlinjNew York, 1993. 10. S. A. Shkarin, On Rolle's theorem in infinite-dimensional Banach spaces, Mat. Zametki 51, No. 3 (1992), 128-136.0022-247X10.1006/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