  The $f$ and $d$ electron density of states of the one-dimensional
Falicov-Kimball model are studied in the weak-coupling limit by exact
diagonalization calculations. The resultant behaviors are used to examine the
$d$-electron gap ($\Delta_{d}$), the $f$-electron gap ($\Delta_{f}$), and the
$fd$-electron gap ($\Delta_{fd}$) as functions of the $f$-level energy $E_f$
and hybridization $V$. It is shown that the spinless Falicov-Kimball model
behaves fully differently for zero and finite hybridization between $f$ and $d$
states. At zero hybridization the energy gaps do not coincide ($\Delta_{d}\neq
\Delta_{f} \neq \Delta_{fd}$), and the activation gap $\Delta_{fd}$ vanishes
discontinuously at some critical value of the $f$-level energy $E_{fc}$. On the
other hand, at finite hybridization all energy gaps coincide and vanish
continuously at the insulator-metal transition point $E_f=E_{fc}$. The
importance of these results for a description of real materials is discussed.
