Clarke's generalized gradient and Edalat's L-derivative

Abstract

Clarke (1973, 1975, 1983) introduced a generalized gradient for real-valued Lipschitz continuous functions on Banach spaces. Edalat (2007, 2008) introduced a so-called L-derivative for real-valued functions and showed that for Lipschitz continuous functions

Clarke's generalized gradient is always contained in this L-derivative and that these two notions coincide if the underlying Banach space is finite dimensional. He asked whether they coincide as well if the Banach space is infinite dimensional. We show that this is the case.