We present a numerical method for solving the time-independent thermal radiative transfer (TRT) equation or the neutron transport (NT) equation when the opacity or cross-section varies rapidly in energy (frequency). The approach is based on a rigorous homogenization of the TRT/NT equation in the energy (frequency) variable. Discretization of the homogenized TRT/NT equation results in a multigroup-type system, and can therefore be solved by standard methods. We demonstrate the accuracy and efficiency of the approach on three model problems. First we consider the Elsasser band model with constant temperature and a small line spacing. Second, we consider a neutron transport application for fast neutrons incident on iron, where the characteristic resonance spacing necessitates about 16,000 energy discretization parameters if Planck-weighted cross sections are used. Third, we consider an atmospheric TRT problem with an opacity corresponding to water vapor. For all three problems, we demonstrate that we can achieve between 0.1 and 1 percent relative error in the solution, and with several orders of magnitude fewer parameters than a standard multigroup formulation with a comparable accuracy.