Non-monotonicity height is an important index to describe the complexity of dynamics for piecewise monotone functions. Although it is used extensively in the theory of iterative roots, its calculation is still difficult especially in the infinite case. In this paper, by introducing the concept of spanning interval, we first present a sufficient condition for piecewise monotone functions to have height infinity and then an algorithm for finding the spanning intervals is given. We further investigate the density of all piecewise monotone functions with infinite and finite height, respectively, and the results indicate the instability of height. At the end of this paper, the variance of height under composition, especially for functions of height 1 and infinity, are also discussed.

非单调高度是描述分段单调函数动力学复杂性的重要指标。尽管它在迭代根理论中得到了广泛的应用，但是它的计算仍然很困难，尤其是在无限情况下。在本文中，通过引入跨度区间的概念，我们首先给出了分段单调函数具有高度无限的充分条件，然后给出了寻找跨度区间的算法。我们进一步研究了分别具有无限和有限高度的所有分段单调函数的密度，结果表明了高度的不稳定性。在本文的最后，还讨论了合成下的高度方差，特别是对于高度1和无穷大的函数。