On functions of two variables equicontinuous in one variable (Q1979052)

For a function \(f:{\mathbb{R}}^2\to {\mathbb{R}}\), sections of \(f\) are the functions \(f_x(t)=f(x,t)\) and \(f^y(t)=f(t,y)\), \(t\in {\mathbb{R}}\). The section \(f_x\) is said to be equicontinuous at a point~\(y\) if for every \(\eta >0\) there is a \(\delta >0\) such that for every point~\(v\) with \(|v-y|<\delta \) and for every \(x\in {\mathbb{R}}\) we have \(|f(x,v)-f(x,y)|<\eta \). Among other results, the following theorem is proved: Let \({\mathcal I}\) and \({\mathcal J}\) be \(\sigma \)-ideals of subsets of \({\mathbb{R}}\) and~\({\mathbb{R}}^2\), respectively, such that the vertical projections of sets not belonging to \({\mathcal J}\) do not belong to~\({\mathcal I}\). If all sections \(f_x\), \(x\in {\mathbb{R}}\), of a function \(f\) are equicontinuous at every point and all sections \(f^y\), \(y\in {\mathbb{R}}\), are \({\mathcal I}\)-almost everywhere continuous, then \(f\) is \({\mathcal J}\)-almost everywhere continuous. Some questions are discussed if \(\sigma \)-ideals in the theorem are replaced with ideals.