Generalizing Generic Differentiability Properties from Convex to Locally Lipschitz Functions

David Preiss proved that every locally Lipschitz function on an open subset of a Banach space which has an equivalent norm Gâteaux (Fréchet) differentiable away from the origin is Gâteaux (Fréchet) differentiable on a dense subset of its domain. It is known that every continuous convex function on an open convex subset of such a space is Gâteaux (Fréchet) differentiable on a residual subset of its domain. We show that for a locally Lipschitz function on a separable Banach space (with separable dual) there are residual subsets which if the function were convex would coincide with its set of points of differentiability. These are the sets where the function is fully intermediately differentiable (fully and uniformly intermediately differentiable) and sets where the subdifferential mapping is weak* (norm) lower semi-continuous. We discuss the role of these sets in generating the subdifferential and present a refinement of Preiss′ result.