Summary: The motion of the impact oscillator subjected to harmonic excitation is usually studied through time integration of the dynamics which generally provides results of poor accuracy. The present paper shows that, by a continuation method, an accurate description of the dynamics is possible and that the stability of multiple impact periodic responses can be analytically studied. Bifurcation diagrams are built showing a typical cascade of subharmonic bifurcation pattern. The Feigenbaum ratio is verified with a high accuracy. The main result is an accurate partitioning of the parameter space in zones of similar behaviour. Zones of chaotic motion and coexisting attractors are identified.