  We show that the energy functional of ensemble Density Functional Theory
(DFT) [Perdew et al., Phys. Rev. Lett. 49, 1691 (1982)] in systems with
attractive interactions is a convex function of the fractional particle number
N and is given by a series of straight lines joining a subset of ground-state
energies. As a consequence the exchange-correlation (XC) potential is not
discontinuous for all N. We highlight the importance of this exact result in
the ensemble-DFT description of the negative-U Anderson model. In the atomic
limit the discontinuity of the XC potential is missing for odd N while for
finite hybridizations the discontinuity at even N is broadened. We demonstrate
that the inclusion of these properties in any approximate XC potential is
crucial to reproduce the characteristic signatures of the charge-Kondo effect
in the conductance and charge susceptibility.
