In this paper we consider the long-term behavior of points in R
under iterations of continuous functions. We show that, given any Cantor set
Ξβ embedded in R, there exists a continuous function
Fβ:RβR such that the points that are bounded under
iterations of Fβ are just those points in Ξβ. In the course of
this, we find a striking similarity between the way in which we construct the
Cantor middle-thirds set, and the way in which we find the points bounded under
iterations of certain continuous functions