On the multiplicity of Dini subgradients in separable spaces Original Research Article

For an extended-valued function on a separable Banach space, the Dini subdifferential is either single-valued or empty on a large subset of the domain, because the set of points at which it contains two or more elements can be covered by a countable family of Lipschitz hypersurfaces. This generalizes a finite-dimensional result of Benoist, and has applications to the Hadamard differentiability of regularly locally convex functions.