Composition of two Riemann integrable functions

A function on a bounded interval is Riemann-integrable iff it is bounded and almost everywhere continuous. So the functions $$ f(x) = \begin{cases} 1 & \text{ for }x \ne 0 \\ 0 & \text{ for } x = 0 \end{cases} \quad \text{ and } \quad g(x) = \begin{cases} 1/q & \text{ for }x=p/q \\ 0 & \text{ for } x \notin \mathbb{Q} \end{cases} $$ are Riemann-integrable over any bounded interval, since $f$ is continuous everywhere except at $0$, and $g$ is continuous at every irrational $x$. (In the definition $x=p/q$ is the unique representation of rational $x$ with $p$ and $q$ relatively prime integers and $q>0$.)

The composition of these functions is $$ f(g(x)) = \begin{cases} 1 & \text{ for }x \in \mathbb{Q} \\ 0 & \text{ for } x \notin \mathbb{Q} \end{cases} $$ which is nowhere continuous, so not Riemann-integrable over any interval.