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.

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Last modified April 8, 2014.