14.4 Discontinuous functions[2DS]

Let be in the following \((X,d)\) a metric space.

Definition 359

[2CX]A set \(E\) is called a \(F_𝜎\) if it is a countable union of closed sets.

(See also exercise 3).

E359

[16N]Note that every open set \(A⊂ X\) nonempty is a \(F_𝜎\) set. (Hint: use 1). Hidden solution: [UNACCESSIBLE UUID ’16P’]

E359

[16Q]Prerequisites:9,3.Given a generic \(f:X→ℝ\), show that the set \(E\) of points where \(f\) is discontinuous is a \(F_𝜎\). Hidden solution: [UNACCESSIBLE UUID ’16R’]

E359

[16S]Prerequisites:359.Difficulty:*.

Suppose \((X,d)\) admits a subset \(D\) that is dense but has empty interior. 1

Given a \(E⊂ X\) which is a \(F_𝜎\), construct a function \(f:X→ℝ\) for which \(E\) is the set of points of discontinuity.

Hidden solution: [UNACCESSIBLE UUID ’16T’]

  1. That is, both \(D\) and the complement \(X⧵ D\) are dense. \(X=ℝ\) meets this requirement, taking as an example \(D=ℚ\).