Compositions of \(\varrho \)-upper continuous functions (Q2826334)

In this paper, some properties of compositions of \(\varrho\)-upper continuous functions are presented. The author shows conditions for a function \(f\: I \to I\) under which \(g \circ f\) is \(\varrho \)-upper continuous for each \(\varrho \)-upper continuous function \(g: I \to \mathbb R\). Also, a small generalization of the Luzin-Menchoff theorem and the Zahorski theorem is discussed.NEWLINENEWLINENEWLINEFirst, some notations are presented. Let \(E\) be a measurable subset of \(\mathbb R\) and let \(x\in \mathbb R\). The numbers NEWLINENEWLINE\[NEWLINE \overline {d}(E,x)=\limsup \limits_{t\rightarrow 0^+ k\rightarrow 0^+}\frac {| E\cap [x-t,x+k]|}{k+t} \quad \text{and} \quad \underline {d}(E,x)=\liminf \limits_{t\rightarrow 0^+ k\rightarrow 0^+}\frac {| E\cap [x-t,x+k]|}{k+t}NEWLINE\]NEWLINENEWLINEare called the upper and lower density of \(E\) at \(x\), respectively.NEWLINENEWLINENEWLINELet \(0< \varrho \leq 1\). We say that \(x\) is a point of \(\varrho \)-type upper density of \(E\) if either \(\overline {d}(E,x)>\varrho \) if \(\varrho <1\) or \(\overline {d}(E,x)=1\) if \(\varrho =1\).NEWLINENEWLINELet \(I\) be an open interval and \(0< \varrho \leq 1\). The function \(f\colon I\to \mathbb R\) is called \(\varrho \)-upper continuous at \(x\in I\) provided that there is a measurable set \(E\subset I\) such that \(x\) is a point of \(\varrho \)-type upper density of \(E\), \(x\in E\) and \(f| E\) is continuous at \(x\). If \(f\) is \(\varrho \)-upper continuous at each point of \(I\) we say that \(f\) is \(\varrho \)-upper continuous. The symbol \(\mathcal{UC}_{\varrho}\) denotes the class of all \(\varrho \)-upper continuous functions defined on \(I\).NEWLINENEWLINENEWLINELet \(\mathcal{F}\) be a family of real functions defined on an open interval \(I\). We put NEWLINE\[NEWLINE\mathcal {IC(F)}=\{f:I\to I:(\forall g\in \mathcal{F})(g\circ f\in \mathcal {F})\}.NEWLINE\]NEWLINE The author proves that a measurable function \(f: I\to I\) belongs to \(\mathcal {IC}(\mathcal{UC}_1)\) if and only if the following condition:NEWLINENEWLINENEWLINELet \(x\in I\) and \(E\subset I\) be a measurable set such that \(f(x)\in E\) and \(\overline {d}(E,f(x))=1\). Then, there exists a measurable set \(F\subset f^{-1}(E)\) for which \(\overline {d}(F,x)=1\) is fulfilled.NEWLINENEWLINENEWLINEMoreover, a measurable function \(f:I\to I\) belongs to \(\mathcal {IC}(\mathcal {UC}_\varrho)\), \(\varrho \in (0,1)\), if and only if the following condition:NEWLINENEWLINENEWLINELet \(x\in I\) and \(E\subset I\) be a measurable set such that \(f(x)\in E\) and \(\overline {d}(E,f(x))>\varrho \). Then, there exists a measurable set \(F\subset f^{-1}(E)\) for which \(\overline {d}(F,x)>\varrho \) is fulfilled.NEWLINENEWLINENEWLINEAlso, the author studies compositions of \(\varrho \)-upper continuous functions with homeomorphisms. The author presents the following generalizations of the Luzin-Menchoff theorem and the Zahorski theorem.NEWLINENEWLINENEWLINETheorem. Let \(E\) be a measurable subset of \(\mathbb R\) and let \(X\) be a closed in \(\mathbb R\) subset of \(E\) such that \(\overline {d}(E,x)> 0\) for each \(x \in X\). Then, there exists a perfect set \(P\) such that \(X\subset P\subset E\) and \(\underline {d}(P,x)=\underline {d}(E,x)\) and \(\overline {d}(P,x)=\overline {d}(E,x)\) for each \(x\in X\).NEWLINENEWLINENEWLINETheorem. Let \(E\) be a set of type \(F_\sigma\) such that \(\overline {d}(E,x)> 0\) for each \(x \in E\). There exists a semi-continuous function \(f:\mathbb R \to \mathbb R\) such that \(0 <f(x)\leq 1\) for \(x\in E\), \(f(x)=0\) for \(x\not \in E\) and \(\underline {d}(t\in \mathbb R: \{| f(x)-f(t)| <\varepsilon \},x)=\underline {d}(E,x)\), \(\overline {d}(t\in \mathbb R: \{| f(x)-f(t)| <\varepsilon \},x)=\overline {d}(E,x)\) for each \(x\in E\) and for each \(\varepsilon >0\).