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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 underiterations of $F^*$ are just those points in $\Lambda^*$. In the course ofthis, we find a striking similarity between the way in which we construct theCantor middle-thirds set, and the way in which we find the points bounded underiterations of certain continuous functions.

Iterated functions and the Cantor set in one dimension