Basic Analysis I — Real Analysis · Problem 3.6.11

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · monotone-functions-continuity

Problem

(Challenging) Find an example of an increasing function

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

that has a discontinuity at each rational number. Then show that the image

f\bigl([0,1]\bigr)

contains no interval. Hint: Enumerate the rational numbers and define the function with a series.