The frequency dispersion of the dynamic conductivity of graphene, of a multilayer graphene, and of IV-VI semiconductors is considered as a function of the temperature and carrier density in the range of frequencies that are higher than the carrier relaxation rate but are lower than the conduction band width. A narrow gap and the linearity of the electron spectrum, which are common features of these materials, are responsible for a singularity of the dielectric function (logarithmic in the real part and step-like in the imaginary part) at the threshold of direct interband transitions and, accordingly, for an anomalously large permittivity in IV-VI semiconductors. The calculated and measured dielectric functions are in a very good agreement. The graphene transmittance in the optical range is frequency-independent and its departure from unity yields the value of the fine structure constant. The difference in dimensionality, which is equal to three for semiconductors and to two for graphene, manifests itself in the different character of plasmons and of electromagnetic waves existing for high doping (or in conditions of the field effect) near the absorption threshold.