Compositions of \(\varrho\)-lower continuous functions (Q2057285)

Let \(E\) be a measurable subset of \(\mathbb{R}\) and let \(x\in\mathbb{R}\). The numbers \[ \underline{d}(E,x)=\liminf_{\substack{t\to0^+ \\ k\to0^+}}\frac{\lambda(E\cap[x-t,x+k])}{k+t} \] and \[ \overline{d}(E,x)=\limsup_{\substack{t\to0^+ \\ k\to0^+}}\frac{\lambda(E\cap[x-t,x+k])}{k+t} \] are called \emph{the lower and the upper density of} \(E\) \emph{at} \(x\), respectively. Let \(0\leq\rho<1\). We say that \(x\) is a point of \(\rho\)\emph{-type lower density of} E if \(\underline{d}(E,x)>\rho\). A real-valued function \(f\) defined on \(I\) is called \begin{itemize} \item \(\rho\)\emph{-lower continuous at} \(x\), provided that for every \(\varepsilon>0\) there is a measurable set \(E\subset\{y\in I\colon|f(x)-f(y)|<\varepsilon\}\) such that \(x\) is a point of \(\rho\)-type lower density of \(E\); \item \(\rho\)\emph{-lower path continuous at} \(x\), provided that there is a measu\-rable set \(E\subset I\) such that \(x\) is a point of \(\rho\)-type lower density of \(E\), \(x\in E\), and \(f_{\arrowvert E}\) is continuous at \(x\). \end{itemize} In the first part of the paper, the authors show that every \(\rho\)-lower continuous function is measurable and give a number of properties and relations between \(\rho\)-lower continuous and \(\rho\)-lower path continuous functions. In the second part of the manuscript, the authors study properties of compositions of \(\rho\)-lower continuous functions. They provide, among others, necessary and sufficient conditions for that the composition of measurable function and \(\rho\)-lower continuous (\(\rho\)-lower path continuous) function is \(\rho\)-lower con\-tinuous (\(\rho\)-lower path continuous). Finally, in the last part of the paper, the authors study properties of compositions of \(\rho\)-lower continuous functions with homeomorphisms.