  We show that in the presence of free charge carriers the definition of the
frequency-dependent dielectric permittivity requires additional regularization.
As an example, the dielectric permittivity of the Drude model is considered and
its time-dependent counterpart is derived and analyzed. The respective electric
displacement cannot be represented in terms of the standard Fourier integral.
The regularization procedure allowing to circumvent these difficulties is
suggested. For purpose of comparison it is shown that the frequency-dependent
dielectric permittivity of insulators satisfies all rigorous mathematical
criteria. This permits us to conclude that in the presence of free charge
carriers the concept of dielectric permittivity is not as well defined as for
insulators and we make a link to widely discussed puzzles in the theory of
thermal Casimir force which might be caused by the use of such kind
permittivities.
