Riemann Integration Problem Help

Question

I've got a question:

Let $f : [a, b] → \mathbb R$ be a function. If $|f|$ is integrable on $[a, b]$, does it follow that f is also integrable on$ [a, b]$?

This is Riemann but not really sure how to answer his question.

Any help will be appreciated

Answer 1

No. Let

$f(x)=1$ if $x \in \mathbb Q$ and $f(x)=-1$ if $x \in\mathbb R \setminus \mathbb Q$

Then $f$ is not integrable on each $[a, b]$, but $|f|$ is.