ABSTRACT
This chapter provides examples which show that in this case the rotation number need not depend continuously on the map. The concept of graph convergence of functions proves to be particularly helpful in investigating the relationship between families of strictly increasing functions and families of left inverse continuous functions which have flats corresponding to the jump discontinuities of the original functions. The examples show that for one parameter families of non-decreasing functions the rotation number need not be a continuous function of the parameter. The chapter shows that if all the functions are continuous, or if all the functions are strictly increasing, then the rotation number is a continuous function of the parameter. If the rotation number of a homeomorphism of the circle is irrational then the limit set of the homeomorphism is either the whole circle or a Cantor subset.
