Smooth Approximation of Lipschitz Maps and Their Subgradients

We derive new representations for the generalised Jacobian of a locally Lipschitz map between finite dimensional real Euclidean spaces as lower limit (i.e., inferior) classical derivative where it exists. The lead to significantly shorter proofs basic properties subgradient and including chain rule. establish that sequence maps converges given in L-topology—that is, weakest refinement sup norm topology on space makes continuous functional—if only if superior directional derivatives vector direction coincides with direction, convergence being uniform all unit vectors. then prove our main result subspace C ∞ is dense equipped L-topology, and, map, we explicitly construct converging allowing global smooth approximation its differential properties. As an application, obtain short proof extension Green’s theorem interval-valued fields. For infinite dimensions, show Banach upper continuous, real-valued separable space, Gateaux differentiable functions such any direction.