Sets of points of discontinuity

Abstract

In order that a subset F F of a topological space coincide with the set of points of discontinuity of a real-valued function on the space, it is necessary that F F be an F σ {F_\sigma } -set devoid of isolated points. It is shown that this condition is also sufficient if the space is “almost-resolvable", and in particular if the space is either separable, first countable, locally compact Hausdorff, or topological linear.