The Differentiability of Real Functions on Normed Linear Space Using Generalized Subgradients*

The modification of the Clarke generalized subdiNerentia1 due to Michel and Penot is a useful tool in determining differentiability properties for certain classes of real functions on a normed linear space. The Glteaux differentiability of any real function can be deduced from the GBteaux differentiability of the norm if the function has a directional derivative which attains a constant related to its generalized directional derivative. For any distance function on a space with uniformly Gateaux differentiable norm. the Clarke and Michel-Penot generalized subdifferentials at points off the set reduce to the same object and this generates a continuity characterization for GLteaux differentiability. However, on a Banach space with rotund dual, the Frechet differentiability of a distance function implies that it is a convex function. A mean value theorem for the modified generalized subdifferential has implications for Glteaux differentiability. b 1987 Academic Press, Inc.