On the Interchange of Subdifferentiation and Epi-Convergence

Given a sequence (ƒn) of nonconvex nondifferentiable functions, or, more generally, a parametrized family of such functions which epi-converges to some function ƒ0 we give conditions in order to have ∂−ƒ0 = limn ∂−ƒn, where ∂−ƒ0 is the Fréchet subdifferential of ƒ0. We also give conditions in order to get inclusons instead of equality. We point out an application to second-order epi-derivatives.