SMOOTH PEANO FUNCTIONS FOR PERFECT SUBSETS OF THE REAL LINE.

In this paper we investigate for which closed subsets P of the real line R there exists a continuous map from P onto P² and, if such a function exists, how smooth can it be. We show that there exists an infinitely many times differentiable function f:R → R² which maps an unbounded perfect set P onto P². At the same time, no continuously differentiable function f:R → R² can map a compact perfect set onto its square. Finally, we show that a disconnected compact perfect set P admits a continuous function from P onto P² if, and only if, P has uncountably many connected components. [ABSTRACT FROM AUTHOR]