The longitudinal dielectric constant is calculated for the extended Bardeen-Cooper-Schrieffer model of a superconductor. An explicit formula is obtained, which can be evaluated numerically. The competition between collective and single-particle effects is pronounced, so that the dielectric finction differs remarkably from the most elementary approximations to it. However, the dielectric function of the superconductor does not differ greatly from that in the normal metal in either the high-frequency or static limit, regardless of the wavelength. This prevents the modifications resulting from superconductivity from being readily observed. In particular, the shift in the static polarizability should cause very small shifts in the phonon speed, so small that no effect on the lattice specific heat should be observed. (auth)