  We present fundamental constraints required for a consistent linear response
theory of fermionic superfluids and address temperatures both above and below
the transition temperature $T_c$. We emphasize two independent constraints, one
associated with gauge invariance (and the related Ward identity) and another
associated with the compressibility sum rule, both of which are satisfied in
strict BCS theory. However, we point out that it is the rare many body theory
which satisfies both of these. Indeed, well studied quantum Hall systems and
random-phase approximations to the electron gas are found to have difficulties
with meeting these constraints. We summarize two distinct theoretical
approaches which are, however, demonstrably compatible with gauge invariance
and the compressibility sum rule. The first of these involves an extension of
BCS theory to a mean field description of the BCS-Bose Einstein condensation
crossover. The second is the simplest Nozieres Schmitt- Rink (NSR) treatment of
pairing correlations in the normal state. As a point of comparison we focus on
the compressibility $\kappa$ of each and contrast the predictions above $T_c$.
We note here that despite the compliance with sum rules, this NSR based scheme
leads to an unphysical divergence in $\kappa$ at the transition. Because of the
delicacy of the various consistency requirements, the results of this paper
suggest that avoiding this divergence may repair one problem while at the same
time introducing others.
