  The interband optical response of a three-dimensional Dirac cone is linear in
photon energy ($\Omega$). Here, we study the evolution of the interband
response within a model Hamiltonian which contains Dirac, Weyl and gapped
semimetal phases. In the pure Dirac case, a single linear dependence is
observed, while in the Weyl phase, we find two quasilinear regions with
different slopes. These regions are also distinct from the large-$\Omega$
dependence. As the boundary between the Weyl (WSM) and gapped phases is
approached, the slope of the low-$\Omega$ response increases, while the
photon-energy range over which it applies decreases. At the phase boundary, a
square root behaviour is obtained which is followed by a gapped response in the
gapped semimetal phase. The density of states parallels these behaviours with
the linear law replaced by quadratic behaviour in the WSM phase and the square
root dependence at the phase boundary changed to $|\omega|^{3/2}$. The optical
spectral weight under the intraband (Drude) response at low temperature ($T$)
and/or small chemical potential ($\mu$) is found to change from $T^2$ ($\mu^2$)
in the WSM phase to $T^{3/2}$ ($|\mu|^{3/2}$) at the phase boundary.
