On the mean value theorem

Abstract

Mean value theorems for nonsmooth functions are presented. Two versions are given, both using the contingent derivative. In. the first one a tangential convexity condition is used. In the second one no convexity assumption is made but the estimate. involves the contingent derivative df (x, b − a) of f at points arbitrarily close to the segment [a, b]. The proof relies on the Bishop-Phelps-Daneš-Ekeland procedure.