On pointwise, discrete and transfinite limits of sequences of closed graph functions (Q1852435)

A function \(f: X\to Y\) (where \(X\) and \(Y\) are topological spaces) has closed graph iff \(G(f)= \{(x,y)\in X\times Y:y= f(x)\}\) is closed in the product topology. \(f\) is the discrete limit of a sequence \(f_n: X\to Y\) iff, for \(x\in X\), there is \(n(x)\in\mathbb{N}\) such that \(f_n(x)= f(x)\) whenever \(n> n(x)\) [see \textit{Á. Császár} and \textit{M. Laczkovich}, Stud. Sci. Math. Hung. 10, 463-472 (1975; Zbl 0405.26006)]. Now let \(X\) be a complete metric space and \(Y=\mathbb{R}\) (with the Euclidean topology). The author proves that, if \(f\) is a pointwise limit of \(f_n\) having closed graph then \(f\) is of the first Baire class; if \(X\) is separable and \(f\) is a discrete limit of \(f_n\) with closed graph then \(f\) is the discrete limit of a sequence of continuous functions; if \(X\) is separable and \(f\) is the limit of a transfinite sequence \(f_\alpha\) \((\alpha< \omega_1)\) of functions with closed graph then there is \(\beta<\omega_1\) such that \(f_\alpha= f\) for \(\alpha>\beta\). If \(X= \mathbb{R}\) and \(G(f)\) is closed in the product of the density topology and the Euclidean topology then \(f\) is measurable; conversely, if \(f\) is measurable then it is the discrete limit of a sequence of functions with closed graph in the above product.