In this paper we discuss second-order properties of the Moreau-Yosida regul
arization F of a piecewise twice continuously differentiable convex functio
n f. We introduce a new constraint qualification in order to prove that the
gradient of F is piecewise continuously differentiable. In addition, we di
scuss conditions, depending on the Hessians of the pieces, that guarantee p
ositive definiteness of the generalized Jacobians of the gradient of F.