Basic Analysis I — Real Analysis · Problem 6.2.17

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · dirichlet-function-pointwise-limit

Problem

(Challenging): The Dirichlet function

f : [0, 1] \to \mathbb{R}

, that is the function such that

f(x) \mathrel{:=} 1

x \in \mathbb{Q}

and

f(x) \mathrel{:=} 0

x \notin \mathbb{Q}

, is not the pointwise limit of continuous functions, although this is difficult to show. Prove, however, that

f

is a pointwise limit of functions that are themselves pointwise limits of continuous functions themselves.