## Real Analysis Exchange

### 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.

#### Article information

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

Dates
First available in Project Euclid: 1 July 2014