  In this paper we consider a one dimensional liner piecewise-smooth
discontinuous map. It is well known that stable periodic orbits exist in this
type of map for a specific parameter region. It is also known that the
corresponding bifurcation phenomena termed as "period adding bifurcation"
exhibits a special structure for such case. In last couple of years, various
authors have analyzed this structure using border collision bifurcation (BCB)
curves and given characterization for various parameter regions. In this paper
we have analyzed a specific parameter space using a different method. We show
that this method enables us to pose some interesting problems like: is there
any relationship between the period of an orbit and the cardinality of orbits?
We prove that such a relationship exists and exactly \phi(n) orbits exist for a
period n where \phi{} is the Euler's number. We propose an algorithm which
gives the location of fixed points of all such \phi(n) distinct orbits and
gives their range of existence with respect to parameters. Further, we show
that using a novel algorithm the computations to find their range of existence
can be reduced.
