Nowhere dense sets and real-valued functions with closed graphs

Closed and nowhere dense subsets which coincide with the points of discontinuity of real-valued functions with a closed graph on spaces which are not necessarily perfectly normal are investigated. Certain Gδ subsets of completely regular and normal spaces are characterized. It is also shown that there exists a countable connected Urysohn space X with the property that no closed and nowhere dense subset of X coincides with the points of discontinuity of a real-valued function on X with a closed graph