The detected thermal contrast is a recently defined figure of merit introduced to describe the overall performance of a detector detecting radiation from a thermal source. We examine the detected thermal contrast for the case where the target emissivity can be assumed to be a function of the temperature and independent of the wavelength within a narrow wavelength interval of interest. Exact expressions are developed to evaluate the thermal contrast detected by both thermal and quantum detectors for focal-plane radiation detecting instruments. Expressions for the thermal contrast of a blackbody, an intrinsic radiative quantity of a body independent of the detection process, and simplified expressions for the detected thermal contrast for target emissivities which are well approximated by the grey body approximation are also given. It is found the contribution in the detected thermal contrast consists of two terms. The first results from changes occurring in the emissivity of a target with temperature while the second results from purely radiative processes. The size of the detected thermal contrast is found to be similar for the two detector types within typical infrared wavelength intervals of interest, contradicting a result previously reported in the literature. The exact results are presented in terms of a polylogarithmic formulation of the problem and extend a number of approximation schemes that have been proposed and developed in the past.