  In this paper we study perturbations of constant cocycles for actions of
higher rank semi-simple algebraic groups and their lattices. Roughly speaking,
for ergodic actions, Zimmer's cocycle superrigidity theorems implies that the
perturbed cocycle is measurably conjugate to a constant cocycle modulo a
compact valued cocycle. The main point of this article is to see that a cocycle
which is a continuous perturbation of a constant cocycle is actually
continuously conjugate back to the original constant cocycle modulo a cocycle
that is continuous and "small".
  We give some applications to perturbations of standard actions of higher rank
semisimple Lie groups and their lattices. Some of the results proven here are
used in our proof of local rigidity for affine and quasi-affine actions of
these groups.
  We also improve and extend the statements and proofs of Zimmer's cocycle
superrigidity.
