  It was found by Perdew, Parr, Levy, and Balduz [Phys. Rev. Lett. {\bf 49},
1691 (1982)] and by Sham and Schl\"uter [Phys. Rev. Lett. {\bf 51}, 1884
(1983)] that the exact Kohn-Sham exchange-correlation potential of an open
system may jump discontinuosly as the particle number crosses an integer, with
important physical consequences. Recently, Sagvolden and Perdew [Phys. Rev. A
{\bf 77}, 012517 (2008)] have analyzed the discontinuity of the
exchange-correlation potential as the particle number crosses one, with an
illustration that uses a model density for the H$^-$ ion. In this work, we
extend their analysis to the case in which the external potential is the simple
harmonic confinement, choosing spring-constant values for which the
two-electron hamiltonian has an analytic solution. This way, we can obtain the
exact, analytic, exchange and correlation potentials for particle number
fluctuating between zero and two, illustrating the discontinuity as the
particle number crosses one without introducing any model or approximation. We
also discuss exchange and correlation separately.
