  We study some special classes of piecewise continuous maps on a finite smooth
partition of a compact manifold and look for invariant measures for such maps.
We show that in the simplest one-dimensional case (so-called interval
translation maps) a Borel probability non-atomic invariant measure exists for
any map. We use this result to demonstrate that any interval translation map
endowed with such a measure is metrically equivalent to an interval exchange
map. Finally, we study the general case of piecewise continuous maps and prove
a simple result on existence of an invariant measure provided all discontinuity
points are wandering.
