Differentiable functions with surjective Clarke Jacobians

Abstract

We construct, for any n,m ∈ N \{0}, a differentiable locally Lipschitz function f : Rn →Rm which is C1 on the complement of an H1-null set E ⊂ Rn and has the property that the range of its limiting Jacobian on E contains the family of all nonempty compact connected sets of (m×n)-matrices. As a consequence, the Clarke Jacobian Jcf is surjective, that is, its range contains every nonempty compact convex subset of (m×n)-matrices. This reveals a significant difference between differentiable functions and C1-functions, since for a C1-function the Clarke Jacobian is always a singleton. As a by-product, we also obtain examples of C1-smooth functions from Rn to Rm (for any n,m ∈ N\{0}) with surjective derivative, that is, Im(Df) = Rm×n.