Nowhere dense sets and real-valued functions with closed graphs (Q1825460)

The author investigates closed and nowhere dense subsets which coincide with the sets of points of discontinuity of closed-graph real-valued functions defined on normal, completely regular or Urysohn spaces. Apparently, constructions of the specific functions fullfilling the prescribed conditions follow an easy pattern (known earlier for perfectly normal (domain) spaces): Let P be a property of a space X and let S be a property of subset \(F\subset X\). Further assume that there is a continuous real-valued function f such that \(f^{-1}(0)=F\) (if P denotes ``normality'' and S denotes \(``G_{\delta}''\) you may quote K. Kuratowski's ``Topology'' monograph, if P stands for ``complete regularity'' and S denotes ``compact \(G_{\delta}''\)- quote L. Gillman and M. Jerison's monograph). Define g: \(R\to R\) by \(g(0)=0\) and \(g(x)=\frac{1}{x}\), if \(x\neq 0\); then take \(h=g\circ f\)- it is easy to show that h has a closed graph. Hence the set of points of discontinuity \(D(h)=F !\) What makes this article really interesting are the proofs of the necessity conditions and non-trivial examples. Analogous problems have been treated by \textit{A. Szymański} and the reviewer [Rend. Circ. Mat. Palermo, II. Ser. 37, No.1, 88-99 (1988; Zbl 0672.54009), here p. 91].