(4.5.5) Helgaker : With regard to contribution (4.5.4), subdifferentiability of F on a dense subset of the N-representable densities I N (namely, on the set of ensemble v-representable densities B N ) follows from the convexity and lower semicontinuity of the universal density functional F. It is a general result of convex analysis that a proper lower semicontinuous convex function (here F) is subdifferentiable on a dense subset (here B N ) of its effective domain (here I N ) and everywhere in the interior of its effective domain (here the empty set since I N has no interior).