Basic Analysis I — Real Analysis · Problem 6.2.16

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · continuous-functions-converging-to-popcorn

Problem

(Challenging): Find a sequence of continuous functions

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

that converge to the popcorn function

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

, that is the function such that

f(p/q) \mathrel{:=} 1/q

(if

p/q

is in lowest terms) and

f(x) \mathrel{:=} 0

x

is not rational (note that

f(0) = f(1) = 1

), see Example 3.2.12. So a pointwise limit of continuous functions can have a dense set of discontinuities.