Measure, Integration & Real Analysis · Problem 6E.7

L^p Spaces — Sheldon Axler

Mathematics — Measure Theory · thomae-function-baire

Problem

(a) Define

f: \mathbf{R} \to \mathbf{R}

as follows:

f(a) = \begin{cases} 0 & \text{if } a \text{ is irrational,} \\ \frac{1}{n} & \text{if } a \text{ is rational and } n \text{ is the smallest positive integer such that } a = \frac{m}{n} \text{ for some integer } m. \end{cases}

At which numbers in

\mathbf{R}

f

continuous? (b) Show that there does not exist a countable collection of open subsets of

\mathbf{R}

whose intersection equals

\mathbf{Q}

. (c) Show that there does not exist a function

f: \mathbf{R} \to \mathbf{R}

such that

f

is continuous at each element of

\mathbf{Q}

and discontinuous at each element of

\mathbf{R} \setminus \mathbf{Q}