The dielectric response of a polar solvent to an ion is analyzed in terms of the bound charge, the net charge that accumulates near the ion as a consequence of the inhomogeneous polarization of the surrounding solvent. We demonstrate that the total bound charge arising in a full molecular treatment is identical to the total bound charge from standard continuum theory. In continuum theory, the bound charge resides in an infinitely thin layer, while in a molecular description the bound charge is spread over a region of finite width. Near simple atomic ions, the width of the bound charge distribution is roughly 1.3 nm. By simulating a sequence of ion charges from 0.1 to 2 e, where e is the magnitude of the electron charge, we analyze the applicability of linear response theory, which has been used by several authors. With increasing charge, the nonlinear response extends to an increasing distance from the ion. However, outside the region containing bound charge, the response is linear and in accord with continuum theory. Previous attempts to assign a dielectric constant to a solvent in the interfacial region are analyzed.