Real Analysis Exchange

Smooth Peano Functions for Perfect Subsets of the Real Line

Krzysztof Chris Ciesielski and Jakub Jasinski

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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.

Article information

Real Anal. Exchange, Volume 39, Number 1 (2013), 57-72.

First available in Project Euclid: 1 July 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 26A30: Singular functions, Cantor functions, functions with other special properties
Secondary: 26B05: Continuity and differentiation questions 58C05: Real-valued functions

Peano curve space filing curve differentiability


Ciesielski, Krzysztof Chris; Jasinski, Jakub. Smooth Peano Functions for Perfect Subsets of the Real Line. Real Anal. Exchange 39 (2013), no. 1, 57--72.

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