  We formulate a continuum linear response theory on the basis of the
Hartree-Fock-Bogoliubov formalism in the coordinate space representation in
order to describe low-lying and high-lying collective excitations which couple
to one-particle and two-particle continuum states. Numerical analysis is done
for the neutron drip-line nucleus $^{24}$O. A low-lying collective mode that
emerges above the continuum threshold with large neutron strength is analyzed.
The collective state is sensitive to the density-dependence of the pairing. The
present theory satisfies accurately the energy weighted sum rule. This is
guaranteed by treating the pairing selfconsistently both in the static HFB and
in the dynamical linear response equation.
