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 O-24. 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 self-consistently both in the static HFB and in the dynamical linear response equation. (C) 2001 Elsevier Science B.V. All rights reserved.