On affinity of Peano type functions

Tomasz Słonka¹

¹ Uniwersytet Śląski 40-007 Katowice, Poland

Colloquium Mathematicum, Tome 127 (2012) no. 2, pp. 233-242.

Résumé

We show that if $n$ is a positive integer and $2^{{\mathrm{\aleph }}_0}\le {\mathrm{\aleph }}_n$, then for every positive integer $m$ and for every real constant $c>0$ there are functions $f_1,\dots ,f_{n+m}:{\mathbb{R}}^n\to \mathbb{R}$ such that $(f_1,\dots ,f_{n+m})({\mathbb{R}}^n)={\mathbb{R}}^{n+m}$ and for every $x\in {\mathbb{R}}^n$ there exists a strictly increasing sequence $(i_1,\dots ,i_n)$ of numbers from $\{1,\dots ,n+m\}$ and a $w\in {\mathbb{Z}}^n$ such that \[ (f_{i_1},\dots,f_{i_n})(y)=y+w \quad \mbox{for } y \in x +(-c,c) \times \mathbb R^{n-1}. \]

Détail

DOI : 10.4064/cm127-2-6

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