  We show that at the level of BCS mean-field theory, the superconducting $T_c$
is always increased in the presence of disorder, regardless of order parameter
symmetry, disorder strength, and spatial dimension. This result reflects the
physics of rare events - formally analogous to the problem of Lifshitz tails in
disordered semiconductors - and arises from considerations of spatially
inhomogeneous solutions of the gap equation. So long as the clean-limit
superconducting coherence length, $\xi_0$, is large compared to disorder
correlation length, $a$, when fluctuations about mean-field theory are
considered, the effects of such rare events are small (typically exponentially
in $[\xi_0/a]^d$); however, when this ratio is $\sim 1$, these considerations
are important. The linearized gap equation is solved numerically for various
disorder ensembles to illustrate this general principle.
