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.

Keywords:

• Contingent derivative
• DINI derivative
• drop
• mean-value theorem
• nonsmooth analysis

AMS 1980 Subject Classification:

• Primary:26 B 05
• Secondary:49 A 27