  We study the spontaneous symmetry breaking (SSB) of a superfluid Bose-Fermi
(BF) mixture in a double-well potential (DWP). The mixture is described by the
Gross-Pitaevskii equation (GPE) for the bosons, coupled to an equation for the
order parameter of the Fermi superfluid, which is derived from the respective
density functional in the unitarity limit (a similar model applies to the BCS
regime too). Straightforward SSB in the degenerate Fermi gas loaded into a DWP
is impossible, as it requires an attractive self-interaction, while the
intrinsic nonlinearity in the Fermi gas is repulsive. Nonetheless, we
demonstrate that the symmetry breaking is possible in the mixture with
attraction between fermions and bosons, like 40K and 87Rb. Numerical results
are represented by dependencies of asymmetry parameters for both components on
particle numbers of the mixture, N_F and N_B, and by phase diagrams in the
(N_F,N_B) plane, which displays regions of symmetric and asymmetric ground
states. The dynamical picture of the SSB, induced by a gradual transformation
of the single-well potential into the DWP, is reported too. An analytical
approximation is proposed for the case when GPE for the boson wave function may
be treated by means of the Thomas-Fermi (TF) approximation. Under a special
linear relation between N_F and N_B, the TF approximation allows us to reduce
the model to a single equation for the fermionic function, which includes
competing repulsive and attractive nonlinear terms. The latter one directly
displays the mechanism of the generation of the effective attraction in the
Fermi superfluid, mediated by the bosonic component of the mixture.
