Riemann integrable functions

Question

If functions $f$ and $g$ are not Riemann integrable when $x ∈ [a,b]$ is then function $fg$ Riemann integrable when $x ∈ [a,b]$? I know that it is possible that function $f+g$ can be Riemann integrable even if $f$ and $g$ are not for example when $f(x)+g(x)=1$. Can I prove $fg$ in the same way or is it not Riemann integrable?

Answer 1

It can be Riemann-integrable. For instance let$$f(x)=\begin{cases}1&\text{ if }x\in\mathbb Q\\0&\text{ otherwise}\end{cases}$$and let$$g(x)=\begin{cases}1&\text{ if }x\notin\mathbb Q\\0&\text{ otherwise.}\end{cases}$$Neither of them is Riemann-integrable, but their product is the null function, which is Riemann-integrable.