  A Borromean nucleus is a bound three-body system which is pairwise unbound
because none of the two-body subsystem interactions are strong enough to bind
them in pairs. As a consequence, the single-particle spectrum of a neutron in
the core of a Borromean nucleus is purely continuum, similarly to the spectrum
of a free neutron, but two valence neutrons are bound up in such a core. Most
of the usual approaches do not use the true continuum to solve the three-body
problem but use a discrete basis, like for example, wave functions in a finite
box. In this paper the proper continuum is used to solve the pairing
Hamiltonian in the continuum spectrum of energy by using the single particle
level density devoid of the free gas. It is shown that the density defined in
this way modulates the pairing in the continuum. The partial-wave occupation
probabilities for the Borromean nuclei $^6$He and $^{11}$Li are calculated as a
function of the pairing strength. While at the threshold strength the
$(s_{1/2})^2$ and $(p_{3/2})^2$ configurations are equally important in $^6$He,
the $(s_{1/2})^2$ configuration is the main one in $^{11}$Li. For very small
strength the $(s_{1/2})^2$ configuration becomes the dominant in both Borromean
nuclei. At the physical strength, the calculated wave function amplitudes show
a good agreement with other methods and experimental data which indicates that
this simple model grasps the essence of the pairing in the continuum.
