  We prove that two smooth families of 2-connected domains in $\cc$ are
smoothly equivalent if they are equivalent under a possibly discontinuous
family of biholomorphisms. We construct, for $m \geq 3$, two smooth families of
smoothly bounded $m$-connected domains in $\cc$, and for $n\geq2$, two families
of strictly pseudoconvex domains in $\cc^n$, that are equivalent under
discontinuous families of biholomorphisms but not under any continuous family
of biholomorphisms. Finally, we give sufficient conditions for the smooth
equivalence of two smooth families of domains.
