In this paper we investigate for which closed subsets P of the real line **R** there exists
a continuous map from P onto P^{2} and, if such a function exists, how smooth can it be.
We show that there exists an infinitely many times differentiable function f:**R**-->**R**^{2}
which maps an unbounded perfect set P onto P^{2}. At the same time, no continuously differentiable function
f:**R**-->**R**^{2}
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 P^{2} if, and only if, P has uncountably many connected components.

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**Last modified April 8, 2014.**