The electronic susceptibility, x(ω, q1, z, z'), of the jellium half-space is evaluated microscopically for finite frequency ω and surface-parallel wavenumber (formula presented) Screening is performed both with and without a self-consistent exchange and correlation potential,(formula presented)whose form is derived from static local density functional theory. The resulting x-functions can be termed local density functional (ldf) and random-phase approximation (rpa) susceptibilities respectively.A q1 integration yields the damping time of an adsorbed oscillating point dipole, which, for separations up to a few angstroms from an aluminium surface, is almost independent of frequency up to around two-thirds of the surface plasmon frequency. The ldf and rpa decay times differ markedly, especially for metals such as sodium with higher rs values, as previously predicted by Liebsch on the basis of a low-frequency expansion. The rpa lifetime for ‘point- dipole’ N, physisorbed on aluminium is in agreement with that deduced from calculations of Eguiluz. Both rpa and ldf lifetimes are, however, substantially longer than that obtained by extrapolation of previous results valid for large dipole-surface separations (i.e. for small q). This in turn means that electron-hole damping of a point dipole is not after all sufficient by itself to explain the lifetime measurements of Avouris, Schmeisser and Demuth. It is nevertheless a substantial contributor to the damping.