Smooth equivalence of deformations of domains in complex euclidean spaces

We prove that two smooth families of 2-connected domains in $\text{cc}$ are smoothly equivalent if they are equivalent under a possibly discontinuous family of biholomorphisms. We construct, for $m\ge 3$, two smooth families of smoothly bounded $m$-connected domains in $\text{cc}$, and for $n\ge 2$, two families of strictly pseudoconvex domains in ${\text{cc}}^n$, 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.