Separately continuous functions: approximations, extensions, and restrictions (Q1415196)

A function \(f\colon X\times Y\to Z\) is separately continuous if all \(x\)-sections and \(y\)-sections of \(f\) are continuous. The authors collect several results concerning separately continuous functions. In particular, they show the following facts. \textbf{(1)}~Every separately continuous function \(f\colon {\mathbb R}\times {\mathbb R}\to {\mathbb R}\) is planar approximable. \textbf{(2)}~For every \(f\colon {\mathbb R}\times {\mathbb R}\to {\mathbb R}\) there exists a \(c\)-dense set \(D\subset {\mathbb R}^2\) such that \(f| D\) is separately continuous. \textbf{(3)}~Every separately continuous real function defined on a subset of the unit square \(I^2\) can be extended to a quasi-continuous function from \(I^2\) to \(\mathbb R\).