A note on feebly continuous functions (Q1032901)

A function \(f:\mathbb R^2\to\mathbb R\) is \textit{feebly continuous} at \((x,y)\) if there are sequences \(x_m\searrow x\) and \(y_n\searrow y\) with \(\lim_{m\to\infty}\lim_{n\to\infty}f(x_m,y_n)=f(x,y)\). In contrast to the situation for functions \(\mathbb R\to\mathbb R\), which must be feebly continuous except possibly at countably many points, an example is given under CH of a function \(\mathbb R^2\to\mathbb R\) which is nowhere feebly continuous. It is also shown that every function \(\mathbb R^2\to\mathbb R\) taking only two values has a point of feeble continuity.