  Through the motivation of the recent discovery of dispersionless regions in
the band structure of the delafossites, a model density of states of free
fermions including a discontinuity as analytical anomaly is studied. The
resulting temperature dependence of the chemical potential is obtained both
exactly and by different approximation schemes which are then discussed
thoroughly. This includes the introduction of an approximation of the
polylogarithm difference which is capable of accessing a parameter range
neither covered by Sommerfeld expansion nor by Boltzmann approximation. It is
found that the Fermi temperature and several other temperature scales may be
very low, giving rise to experimentally observable behaviours differing from
the one described by Fermi liquid theory. In particular, two kinds of apparent
Fermi liquid behaviour emerge at intermediate temperatures. This behaviour is
related to recent transport data reported for CuCr(1-x)MgxO2 [A. Maignan et.
al, Solid State Commun. 149, 962 (2009)] and CuRh(1-x)MgxO2 [A. Maignan et. al,
Phys. Rev. B 80, 115103 (2009)] by means of the temperature independent
correlation functions ratio approximation. In this way an effective density of
states as well as the effective charge carrier density of these materials are
determined. Furthermore, conclusions about the specific heat of the latter
material are drawn which presents particular effects of the analytical anomaly.
