In this work we investigate a piecewise-linear dis- continuous map defined on three partitions. This map denoted as truncated tent map was specifically constructed in such a way that it shows a similar dynamic behavior like a piecewise- linear continuous map with an additional square root term with positive sign derived from the time continuous model of an impact oscillator. Since the square root part makes analytical studies complicated, the constructed map serves us as an easy to investigate test system. The stable periodic dynamics of this piecewise-linear map was found to be organized by a heretofore unexplained bifurcation structure in an extended region in parameter space, which superficially resembles period adding. This work explains the stable periodic behavior of the truncated tent map and the new bifurcation structure.