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Within exact electron density-functional theory, we investigate Kohn-Sham(KS) potentials, orbital energies, and non-interacting kinetic energies of thefractional ions of Li, C and F. We use quantum Monte Carlo densities as input,which are then fitted, interpolated at non-integer electron numbers $N$, andinverted to produce accurate KS potentials $v_s^N(r)$. We study the dependenceof the KS potential on $N$, and in particular we numerically reproduce thetheoretically predicted spatially constant discontinuity of $v_s^N(r)$ as $N$passes through an integer. We further show that, for all the cases considered,the inner orbital energies and the non-interacting kinetic energy are nearlypiecewise linear functions of $N$. This leads us to propose a simpleapproximation of the KS potential $v_s^N(r)$ at any fractional electron number$N$ which uses only quantities of the systems with the adjacent integerelectron numbers.

Kohn-Sham potentials in exact density-functional theory at non-integer electron numbers