Generalized directional derivatives for locally Lipschitz functions which satisfy Leibniz rule

In this paper a concept of a generalized directional derivative, which satisfies Leibniz rule is proposed for locally Lipschitz functions, defined on an open subset of a Banach space. Although Leibniz rule is of less importance for a subdifferential calculus, it is of course of some theoretical interest to know about the existence of generalized directional derivatives which satisfy Leibniz rule. The proposed concept of generalized directional derivatives is adopted from the work of D. R. Sherbert (1964) who determined all point derivations for the Banach algebra of Lipschitz functions over a complete metric space.