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.
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