  An exact solution is presented for the frequency-dependent charge
susceptibility of the spinless Falicov-Kimball model by using dynamical
mean-field theory. We develop a nontrivial application of the Baym-Kadanoff
"conserving approximation" formalism to exactly determine the
frequency-dependent vertex function (which turns out to assume a particularly
simple form). We show how the static and dynamic susceptibilities are decoupled
in this model and how the dynamic susceptibility generically does not show any
signal of the low-temperature charge-density-wave phase transition. We also
examine the temperature evolution of the dynamic charge susceptibility for the
special case of half-filling.
