On sums and products of extendable functions (Q5932436)

The authors study the maximal additive, multiplicative, lattice-like families for the class of all extendable functions. The main results are the following. NEWLINENEWLINENEWLINETheorem~ 2.1 NEWLINE\[NEWLINE\mathcal M _a(\text{Ext}(\mathbf R,\mathbf R))= \text{C}(\mathbf R,\mathbf R)=\mathcal M _l( \text{Ext}(\mathbf R,\mathbf R)).NEWLINE\]NEWLINE NEWLINENEWLINENEWLINETheorem~ 2.2 NEWLINE\[NEWLINE\mathcal M _m(\text{Ext}(\mathbf R,\mathbf R))= \text{M}.NEWLINE\]NEWLINE NEWLINENEWLINENEWLINETheorem~ 3.1 NEWLINE\[NEWLINE\mathcal M _a(\text{PC}(\mathbf R^2,\mathbf R))= \text{C}(\mathbf R^2,\mathbf R)=\mathcal M _l( \text{PC}(\mathbf R^2,\mathbf R)).NEWLINE\]NEWLINE NEWLINENEWLINENEWLINEHere for topological spaces \(X,Y\): NEWLINENEWLINENEWLINE\(\text{PC}(X,Y)\) -- the class of all peripherally continuous functions. NEWLINENEWLINENEWLINE\(\text{Ext}(X,Y)\) -- the family of extendable functions. NEWLINENEWLINENEWLINE\(\text{M}\) -- the class of all functions \(f:\mathbf R\rightarrow \mathbf R\) such that if \(x_0\) is a right (left) point of discontinuity of \(f\), then \(f(x_0)=0\) and there is a sequence \((x_n)\) converging to \(x_0\) such that \(x_n>x_0\) (\(x_n<x_0\)) and \(f(x_n)=0\). NEWLINENEWLINENEWLINEFor a class \(\mathcal X\) of real functions: NEWLINENEWLINENEWLINEThe maximal additive (multiplicative, lattice-like) class for \(\mathcal X\) is defined to be the class of all \(f\in \mathcal X\) for which \(f+g\in \mathcal X\) (\(fg\in \mathcal X\), \(\max (f,g)\in \mathcal X\), \(\min (f,g)\in \mathcal X\), respectively) whenever \(g\in \mathcal X\). The respective classes are denoted by \(\mathcal M_a(\mathcal X)\), \(\mathcal M_m(\mathcal X)\), \(\mathcal M_l(\mathcal X)\).