  Capturing the discontinuous shift by $\Delta$ in the exact
exchange-correlation (xc) potential is the standard proposal for calculating
the fundamental gap, $E_\mathrm{g}$, from the Kohn-Sham (KS) gap,
$\varepsilon_\mathrm{g}$, within KS density functional theory (DFT), as
$E_\mathrm{g} = \varepsilon_\mathrm{g} + \Delta$, yet this discontinuity is
absent from existing approximations. The '$N$-centered' formulation of ensemble
DFT artificially maintains a total electron number, $N$, in order to yield
$E_\mathrm{g}$ not through a discontinuous shift in the xc potential but via
the ensemble-weight derivative of the xc energy. Within the $N$-centered
approach we calculate exact xc potentials for a one-dimensional finite system
and show analytically that $\Delta$ can in fact be interpreted as a
discontinuous shift in the exact $N$-centered ensemble xc potential, thereby
extending to charged excitations an exact property of uncharged excitations. We
show that applying the Levy-Zahariev 'shift-in-potential' procedure in this
context relocates the discontinuous shift to the unimportant periphery of the
system, so that the exact xc potential in effect is free of discontinuities and
thus the inability of a local functional to capture discontinuous behavior is
inconsequential.
