On affinity of Peano type functions

Volume 127 / 2012

Tomasz Słonka Colloquium Mathematicum 127 (2012), 233-242 MSC: Primary 26B99. DOI: 10.4064/cm127-2-6


We show that if $n$ is a positive integer and $2^{\aleph_0} \leq \aleph_n$, then for every positive integer $m$ and for every real constant $c>0$ there are functions $f_1,\dots,f_{n+m}\colon \mathbb R^n \rightarrow \mathbb R$ 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 \[ (f_{i_1},\dots,f_{i_n})(y)=y+w \quad \mbox{for } y \in x +(-c,c) \times \mathbb R^{n-1}. \]


  • Tomasz SłonkaUniwersytet Śląski
    40-007 Katowice, Poland

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