functions that are Riemann Integrable but its absolute value is not Riemann Integrable?

Question

can anyone give me an example of functions that are Riemann Integrable on $[0, +\infty]$ but its absolute value is not Riemann Integrable on $[0, +\infty]$? Thanks!

Answer 1

It is not possible when considering a Riemann integrable function on a bounded interval.

If $f$ is Riemann integrable then there is a partition $P$ for any $\epsilon > 0$ such that

$$U(P,|f|) - L(P,|f|) \leqslant U(P,f) - L(P,f) < \epsilon,$$

which implies that $|f|$ is Riemann integrable.

This result follows from the reverse triangle inequality $|\,|f(x| - |f(y)|\,| \leqslant |f(x) - f(y)|,$ implying for any partition interval $I$,

$$\sup_{x \in I}|f(x)| - \inf_{x \in I}|f(x)| \leqslant \sup_{x \in I}f(x) - \inf_{x \in I}f(x) $$

Edited Question

The term Riemann integrable is not relevant for the unbounded interval $[0, \infty)$. The usual example of $\sin x / x$ now gives $f$ where the improper integral exists, but fails to converge to a finite value for $|f|$.