The mechanism of period-adding is found in a piecewise linear map with a gap. Both end points of each phase-locking plateau in the staircase are discussed. Our results show that the reason of period-adding phenomenon is the border collision bifurcation at two boundaries of the forbidden area with critical parameters, meanwhile the orbits can iterate to another region and still retains the original region, and then it forms the period-adding sequences. Finally the analytical expressions of the both end points of each phase-locking plateau in the staircase are given.

discontinuity; phase-locking; period-adding