  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 $\Gamma/U$, where $\Gamma$
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.
