Scalar-tensor cosmology at the general relativity limit: Jordan vs Einstein frame

We consider the correspondence between the Jordan frame and the Einstein frame descriptions of scalar-tensor theory of gravitation. We argue that since the redefinition of the scalar field is not differentiable at the limit of general relativity the correspondence between the two frames is lost at this limit. To clarify the situation we analyse the dynamics of the scalar field in different frames for two distinct scalar-tensor cosmologies with specific coupling functions and demonstrate that the corresponding scalar field phase portraits are not equivalent for regions containing the general relativity limit. Therefore the answer to the question whether general relativity is an attractor for the theory depends on the choice of the frame.