How does the piecewise-linearity requirement for the density affect quantities in the Kohn-Sham system?

Kohn-Sham (KS) density functional theory (DFT) is an extremely popular, in-principle exact method, which can describe any many-electron system by introducing an auxiliary system of non-interacting electrons with the same density. When the number of electrons, N, changes continuously, taking on both integer and fractional values, the density has to be piecewise-linear, with respect to N. In this contribution, I explore how the piecewise-linearity property of the exact interacting density is reflected in the KS system. In particular, since the KS potential, v_KS^(N)(r), is N-dependent, so are the KS eigenvalues, orbitals and other resultant quantities. But if the density constructed within the KS system must be strictly linear in N, how does this restrict the N-dependence of other KS quantities? To make progress in this question, I suggest to express KS quantities using the two-point Taylor expansion in N and find how the expansion coefficients are restricted by the piecewise-linearity requirement. I focus on the total electron density, the Kohn-Sham sub-densities and the highest occupied (HOMO) orbital density. In addition to exact analytical results, common approximations for the HOMO, namely the frozen and the linear regimes, are analyzed. A numerical investigation using various exchange-correlation approximations is performed to test the analytical findings. The outcomes of this work will help to remove density-driven errors in DFT calculations for open systems and ensembles.