Open Access
2013/2014 Smooth Peano Functions for Perfect Subsets of the Real Line
Krzysztof Chris Ciesielski, Jakub Jasinski
Real Anal. Exchange 39(1): 57-72 (2013/2014).


In this paper we investigate for which closed subsets \(P\) of the real line \(\mathbb{R}\) 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 \(f\colon\mathbb{R}\to\mathbb{R}^2\) which maps an unbounded perfect set \(P\) onto \(P^2\). At the same time, no continuously differentiable function \(f\colon\mathbb{R}\to\mathbb{R}^2\) 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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Krzysztof Chris Ciesielski. Jakub Jasinski. "Smooth Peano Functions for Perfect Subsets of the Real Line." Real Anal. Exchange 39 (1) 57 - 72, 2013/2014.


Published: 2013/2014
First available in Project Euclid: 1 July 2014

zbMATH: 1301.26009
MathSciNet: MR3261899

Primary: 26A30
Secondary: 26B05 , 58C05

Keywords: differentiability , Peano curve , space filing curve

Rights: Copyright © 2013 Michigan State University Press

Vol.39 • No. 1 • 2013/2014
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