We clarify an important aspect of density functional theories, the broadening of the derivative discontinuity (DD) in a quantum system, with fluctuating particle number. Our focus is on a correlated model system, the single level quantum dot in the regime of the Coulomb blockade. We find that the DD-broadening is controlled by the small parameter Γ/U , where Γ is the level broadening due to contacting and U is a measure of the charging energy. Our analysis suggests, that Kondoesque fluctuations have a tendency to increase the DD-broadening, in our model by a factor of two.PACS numbers: 05.60.Gg, 85.65.+h Introduction. Over the years the density functional theory (DFT) developed into an important tool to study transport properties of nano-systems and single molecules. 1-5 This development occured despite of the fact, that often the results are quantitatively sensitive to the approximations made for the exchange correlation (XC) functional, V XC [n], underlying such calculations. 6-10 In principle, dc-transport calculations should combine either long-time evolution of wavepackets or a KS-based quasistationary formalism, in both cases with special dynamical XC-functionals. 9,11,12 In practice, the available ground state functionals are being used.The neglect of dynamical correlations for simulation of dc-transport was justfied for spinless systems for which a Friedel-sum rule holds. 13 In such systems, approximations to the XC-potential of the ground state introduce the largest numerical error in the regime of Coulomb blockade (CB) where the system ("quantum dot", QD) is only weakly coupled to the electronic reservoirs and the filling is close to an integer. Its signature is an addition energy, U , that largely exceeds the (single particle) level spacing of the QD.In this Letter we exploit the observation that CB is intrinsically an equilibrium phenomenon even though it is mostly discussed in its effect on transport measurements 15 ; in a broader context it is a typical manifestation of the reduced compressibility, dn(r)/dµ, of repulsively interacting fermion gases. Therefore, it has a reincarnation in XC-functionals of DFT where it appears as the derivative discontinuity (DD).Starting from the seminal work by Perdew et al. 16 , the DD was almost exclusively discussed in the limiting case of decoupled quantum dots, i.e. closed systems. There, the XC-functional jumps discontinuously when tuning the particle number, N , of a closed system in its ground state through an integer value