Using a density-functional approach, we study the dielectric permittivity across interfaces at the atomic scale. Focusing on the static and high-frequency permittivities of SiO2 films on silicon, for oxide thicknesses from 12 Angstrom down to the atomic scale, we find a departure from bulk values in accord with experiment. A classical three-layer model accounts for the calculated permittivities and is supported by the microscopic polarization profile across the interface. The local screening varies on length scales corresponding to first-neighbor distances, indicating that the dielectric transition is governed by the chemical grading. Silicon-induced gap states are shown to play a minor role.