  We show that the response to an electric field, in models of the Integral
Quantum Hall effect, is analytic in the field and has isolated essential
singularity at zero field. We also study the breakdown of Chern numbers
associated with the response of Floquet states. We argue, and give evidence,
that the breakdown of Chern numbers in Floquet states is a discontinuous
transition at zero field. This follows from an observation, of independent
interest, that Chern numbers for finite dimensional Floquet operators are
generically zero. These results rule out the possibility that the breakdown of
the Hall conductance is a phase transition at finite fields for a large class
of models.
