The first conceptual development of the chemical potential µ was based on the assumption that the fundamental functional E[N, v(r)] is differentiable everywhere for both variables. Subsequently, the exact piecewise linear conditions at fractional particle numbers were established originally by Perdew, Parr, Levy, and Baldus (PPLB) based on grand canonical ensembles at zero temperature and later by Yang, Zhang, and Ayers based on pure states with degeneracy. The piecewise linearity of E[N, v(r)] with respect to N means that the derivatives at integer electron numbers are discontinuous. In particular, at a given integer N, the chemical poten-