The dynamical density-response function of a periodically modulated two-dimensional electron gas under a perpendicular magnetic field is calculated within the random-phase approximation. It is found that the response function is not only broadened by the periodic potential, it also contains a series of subsingularities at the band edges. The maximum number of subsingularities is 2nmax, where nmax is the number of occupied Landau levels (the last level may be partially filled). It is further proposed that light-scattering or electromagnetic absorption experiments can be performed to study these predicted structures. © 1995 The American Physical Society.