EQUICALMNESS AND EPIDERIVATIVES THAT ARE POINTWISE LIMITS

Recently, Moussaoui and Seeger (Ref. 1) studied the monotonicity of fi rst-order and second-order difference quotients with primary goal the simplification of epilimits. It is well known that epilimits (lim inf and lim sup) can be written as pointwise limits in the case of a seque nce of functions that is equi-lsc. In this paper, we introduce equical mness as a condition that guarantees equi-lsc, and our primary goal is to give conditions that guarantee that first-order and second-order d ifference quotients are equicalm. We show that a piecewise-C-1 functio n f with convex domain is epidifferentiable at any point (x) over bar of its domain. We also show that a convex piecewise C-2-function (poly hedral pieces) is twice epidifferentiable. We thus obtain a modest ext ension of the Rockafellar result concerning the epidifferentiability o f piecewise linear-quadratic convex functions.