Directional derivatives of difference functions of nonsmooth max-functions

In this paper, we give an upper estimate for the Clarke-Rockafellar directional derivatives of a function of the form f − g, where f, g are max-functions defined by locally Lipschitz but not necessarily differentiable functions on a closed convex set in a Euclidean space. As an application, we give a sufficient condition for f − g to have an error bound.