We study the spectral properties of the charge-density-wave (CDW) phase of the half-filled spinless Falicov-Kimball model within the framework of the dynamical mean field theory. We present detailed results for the spectral function in the CDW phase as a function of temperature and U. We show how the proximity of the non-Fermi liquid phase affects the CDW phase, and show that there is a region in the phase diagram where we get a CDW phase without a gap in the spectral function. This is a radical deviation from the mean-field prediction where the gap is proportional to the order parameter.