Density continuous transformations on ℝ2

Abstract

In the paper we study transformations from ${ℝ}^2$ into $ℝ$ and from ${ℝ}^2$ into ${ℝ}^2$ continuous with respect to different density topologies on the domain and range. In the former case the range will always be equipped with the one-dimensional density topology and the domain with either ordinary density or strong density topology. In the later case the domain and the range will be equipped simultaneously with the ordinary density or strong density topology. We will investigate the relations between these classes and with the class of ordinary continuous transformations. We will also examine relation between the (strong) density continuity of ${ℝ}^2\to ℝ$ and of $f(\cdot ,y),f(x,\cdot )$. Similar question will be considered for transformations $F=(f,g):{ℝ}^2\to {ℝ}^2$ and their coordinate functions $f$ and $g$.