Basic Analysis I — Real Analysis · Problem 3.6.14

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · monotone-functions-continuity

Problem

Prove that the Dirichlet function

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

, defined by

f(x) \coloneq 1

x

is rational and

f(x) \coloneq 0

otherwise, cannot be written as a difference of two increasing functions. That is, there do not exist increasing

g

and

h

such that

f(x) = g(x) - h(x)