According to a well-known result the collection
of all ω-limit sets of a continuous map
of the interval equipped with the Hausdorff
metric is a compact metric space. In this paper
a similar result is proved for some piecewise
continuous maps with finitely many points of
discontinuity. We also show that ω-limit
sets of these maps are locally expanding,
another property known for continuous maps.
However, contrary to the situation for
continuous maps, there are piecewise continuous
maps having locally expanding sets which are not
ω-limit sets. A condition implying that a
locally expanding set is an ω-limit set
is presented.