This work develops the semiclassical dielectric theory of energy loss, including retardation effects, for electrons travelling normal to a stratified slab. Starting from a transfer matrix formulation and using computer algebra, we obtain closed formulae for the dispersion relation, the Hertz vector and the energy-loss spectrum, valid for any finite number of layers. Our results are applied to multilayers of Al/Al₂O₃ and are used to investigate the validity of modelling a single interface by a sharp discontinuity in the dielectric function.