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J. Math. Anal. Appl. 283 (2003) 180191 www.elsevier.com/locate/jmaa

Summary: J. Math. Anal. Appl. 283 (2003) 180­191
Approximate Rolle's theorems for the proximal
subgradient and the generalized gradient
Daniel Azagra,1
Juan Ferrera,2
and Fernando López-Mesas ,3
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense,
28040 Madrid, Spain
Received 1 April 2002
Submitted by K. Jarosz
We establish approximate Rolle's theorems for the proximal subgradient and for the generalized
gradient. We also show that an exact Rolle's theorem for the generalized gradient is completely false
in all infinite-dimensional Banach spaces (even when they do not possess smooth bump functions).
2003 Elsevier Inc. All rights reserved.
Keywords: Rolle's theorem; Proximal subgradient; Generalized gradient
1. Introduction
Rolle's theorem in finite-dimensional spaces states that, for every bounded open sub-
set U of Rn and for every continuous function f :U R such that f is differentiable


Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid


Collections: Mathematics