Abstract: As nonlinear operation, iteration usually amplifies the nonlinearity of nonlinear function. Polygonal function with a single vertex is the simplest nonlinear function. As the function values can intersect in different subintervals under iteration, the changes of polylines are most complicated. By using the monotonicity of the polygonal function in subintervals, the sufficient conditions are put forth respectively for cases when the number of vertices is invariant, when the number of vertices is a positive integer and when the number is positive infinity.