  We investigate the phenomenon of fermionic pairing with mismatched Fermi
surfaces in a two-species system in the presence of Feshbach resonance, where
the resonantly-paired fermions combine to form bosonic molecules. We observe
that the Feshbach parameters control the critical temperature of the gapped BCS
superfluid state, and also determine the range over which a gapless breached
pair state may exist. Demanding the positivity of the superfluid density, it is
shown that although a breached pair state with two Fermi surfaces is always
unstable, its single Fermi-surface counterpart can be stable if the chemical
potentials of the two pairing species have opposite signs. This condition is
satisfied only over a narrow region in the BEC side, characterized by an upper
and a lower limit for the magnetic field. We estimate these limits for a
mixture of two hyperfine states of $^6$Li using recent experimental data.
