  In this paper we consider the long-term behavior of points in ${\mathbb R}$
under iterations of continuous functions. We show that, given any Cantor set
$\Lambda^*$ embedded in ${\mathbb R}$, there exists a continuous function
$F^*:{\mathbb R}\to{\mathbb R}$ such that the points that are bounded under
iterations of $F^*$ are just those points in $\Lambda^*$. 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.
