Abstract: For a kind of piecewise map, its nonsmoothness with the parameter z is discussed, and the conditions for border-collision bifurcation and smooth flip/fold bifurcation are derived. These bifurcation conditions are verified by numerical simulation, and the period-adding bifurcation scenario is a general phenomenon for non-smooth maps. The bifurcation scenario is rooted on the interrelation between border-collision bifurcation and smooth flip/fold bifurcation. The parameter z has important effect on the bifurcation scenario, but the parameter has a minor effect, so the analysis γ on bifurcation scenarios with parameters can be simplified by giving γ=0.