  We study the near action of the group PC of piecewise continuous
self-transformations of the circle. Elements of this group are only defined
modulo indeterminacy on a finite subset, which raises the question of
realizability: a subgroup of PC is said to be realizable if it can be lifted to
a group of permutations of the circle. We show that every finitely generated
abelian subgroup of PC is realizable. We show that this is not true for
arbitrary subgroups, by exhibiting a non-realizable finitely generated subgroup
of the group of interval exchanges with flips. The group of (oriented) interval
exchanges is obviously realizable (choosing the unique left-continuous
representative). We show that it has only two realizations (up to conjugation
by a finitely supported permutation): the left and right-continuous ones.
