Sets of points of discontinuity

Abstract

In order that a subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a topological space coincide with the set of points of discontinuity of a real-valued function on the space, it is necessary that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F"> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding="application/x-tex">F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript sigma"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mi> σ </mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{F_\sigma }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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.