  We consider the deformation of a discontinuous group acting on the Euclidean
space by affine transformations.
  A distinguished feature here is that even a `small' deformation of a discrete
subgroup may destroy proper discontinuity of its action. In order to understand
the local structure of the deformation space of discontinuous groups, we
introduce the concepts from a group theoretic perspective, and focus on
`stability' and `local rigidity' of discontinuous groups.
  As a test case, we give an explicit description of the deformation space of
Z^k acting properly discontinuously on R^{k+1} by affine nilpotent
transformations.
  Our method uses an idea of `continuous analogue' and relies on the criterion
of proper actions on nilmanifolds.
