  We address the question whether a graviton could have a small nonzero mass.
The issue is subtle for two reasons: there is a discontinuity in the mass in
the lowest tree-level approximation, and, moreover, the nonlinear
four-dimensional theory of a massive graviton is not defined unambiguously.
First, we reiterate the old argument that for the vanishing graviton mass the
lowest tree-level approximation breaks down since the higher order corrections
are singular in the graviton mass. However, there exist nonperturbative
solutions which correspond to the summation of the singular terms and these
solutions are continuous in the graviton mass. Furthermore, we study a
completely nonlinear and generally covariant five-dimensional model which
mimics the properties of the four-dimensional theory of massive gravity. We
show that the exact solutions of the model are continuous in the mass, yet the
perturbative expansion exhibits the discontinuity in the leading order and the
singularities in higher orders as in the four-dimensional case. Based on exact
cosmological solutions of the model we argue that the helicity-zero graviton
state which is responsible for the perturbative discontinuity decouples from
the matter in the limit of vanishing graviton mass in the full classical
theory.
