  We consider first generation scalar-tensor theories of gravitation in a
completely generic form, keeping the transformation functions of the local
rescaling of the metric and the scalar field redefinition explicitly distinct
from the coupling functions in the action. It is well known that in the Jordan
frame Brans-Dicke type parametrization the diverging kinetic coupling function
$\omega \rightarrow \infty$ can lead to the general relativity regime, however
then the transformation functions to other parametrizations typically become
singular, possibly spoiling the correspondence between different
parametrizations. We give a detailed analysis of the transformation properties
of the field equations with arbitrary metric and also in the Friedmann
cosmology, and provide sufficient conditions under which the correspondence
between different parametrizations is retained, even if the transformation is
singular. It is interesting to witness the invariance of the notion of the
general relativity regime and the correspondence of the perturbed cosmological
equations as well as their solutions in different parametrizations, despite the
fact that in some cases the perturbed equation turns out to be linear in one
parametrization and nonlinear in some other.
