  Standard approximations for the exchange-correlation functional are known to
deviate from linear dependence of the energy on the electron and spin numbers
(in -space). Violation of this flat-plane condition underlies the failure of
all known approximate functionals to describe band gaps in strongly correlated
systems. Then crucial for further development of functionals is recognition of
the behavior of the energy as a function of, and its derivatives (derivatives
discontinuity pattern at 0K limit). In this Letter, the energy derivatives are
analyzed thoroughly. It is shown that apart from the well-known type of
discontinuity pattern, three other patterns are possible in the vicinity of the
singlet state point. The zero temperature limits of energy derivatives at this
point are derived, found different for various directions. Existence of all
discontinuity patterns is illustrated on the example of diatomic molecules set.
