On the set of points of discontinuity for functions with closed graphs.(English)Zbl 0581.54008

For two topological spaces X and Y and any function $$f: X\to Y$$ let $$D_ f$$ denote the set of all points of discontinuity of f. This paper characterizes the set of all points of discontinuity of a real-valued function with closed graph defined on a perfectly normal space X, extending results of I. Baggs [Proc. Am. Math. Soc. 43, 439-442 (1974; Zbl 0281.54005)]. The main results are the following: Theorem 1. Let X be a perfectly normal topological space. Then $$A\subset X$$ is closed and of the first category in X if and only if there exists a real- valued function f with closed graph defined on X such that $$D_ f=A$$. Theorem 2. Let X be a Baire metric space. Then $$F\subset X$$ is closed and nowhere dense in X if and only if there exists a real-valued function f with closed graph defined on X such that $$D_ f=F$$. An example is given showing that the assumption ”X is perfectly normal” in Theorem 1 cannot be replaced by the assumption ”X is normal”.
Reviewer: Y.S.Yang

MSC:

 54C10 Special maps on topological spaces (open, closed, perfect, etc.)

Zbl 0281.54005
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