Smooth Peano Functions for Perfect Subsets of the Real Line

Abstract

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.