  Let $f$ be a piecewise continuous map on the interval with at most a finite
number of turning points. In this paper we study some basic properties about
this class of functions and show its main difference from the continuous case.
We define and study the notion of closed structure, which can be seen as an
extension of the notion of periodic orbit. Moreover, we also study the periodic
orbits which are away from the discontinuities of $f$, extending the notion of
trapped and free periodic orbits.
