Question

Show using an example 𝑓 and 𝑔 are not integrable on [𝑎, 𝑏], but 𝑓𝑔 may be integrable on [𝑎, 𝑏].

Answer 1

Take the two modified Dirichlet function $\delta_1(x)$ and $\delta_2(x)=\delta_1(x)$, where

$\delta_1(x)=\begin{cases} 1,\ x\in\mathbb{R}\setminus\mathbb{Q}\\ -1,\ x\in\mathbb{Q}\end{cases}$

By Darboux criteria they are not integrable on $[0,1]$ , but $\delta_1\delta_2\equiv 1$ is integrable on $[0,1]$