Let an elementary function be a function which is made up of sums, products, and compositions, of finitely many polynomial, rational, trigonometric, and exponential functions, and of their inverses.
I'm wondering whether there exists an elementary function which is not Riemann integrable, but not due to it being unbounded? That is, does there exist an elementary $f(x)$ such that it is bounded on some finite interval $[a,b]$ and yet its Riemann integral $\int_a^b f(x)\,dx$ doesn't exist? I am allowing for $f(x)$ to be undefined in some of the points of $[a,b]$.
My thoughts so far:
From Lebesgue's criterion for Riemann integrability, in order for a bounded $f(x)$ to be integrable on $[a,b]$ its points of discontinuity must be of measure zero. So we are looking for an elementary $f(x)$ with infinitely many points of discontinuity, in fact uncountably many. These points of discontinuity mustn't be asymptotes, because we're looking for bounded $f(x)$.
One way to get an elementary function with a jump discontinuity is to define e.g. $f(x)=\frac{|x|}{x}$ (this indeed fits the definition of an elementary function since we can write $|x|=\sqrt{x^2}$). I thought of somehow composing this with a periodic function to get an infinite number of such jump discontinuities in a finite interval, but this would still give us only a countable number of such points (and hence still have measure zero).
Is there an example of a (bounded) elementary function which is not Riemann integrable? Is there a reason for such functions to never exist?