The confinement of a superconductor in a thin film changes its Fermi-level density of states and is expected to change its critical temperature Tc. Previous calculations have reported large discontinuities of Tc when the chemical potential coincides with a subband edge. By solving the BCS gap equation exactly, we show that such discontinuities are artifacts and that Tc is a continuous function of the film thickness. We also find that Tc is reduced in thin films compared with the bulk if the confinement potential is lower than a critical value, while for stronger confinement Tc increases with decreasing film thickness, reaches a maximum, and eventually drops to zero. Our numerical results are supported by several exact solutions. We finally interpret experimental data for ultrathin lead thin films in terms of a thickness-dependent effective mass.