This is a question from a past qualifying exam, which I am studying for. The question has been asked before here, and has an answer, but the answer uses Lebesgue's criterion for Riemann integrability, which is disallowed on the exam. Is there a more elementary way to solve this question?
Let $f: [0,1] \to \mathbb{R}$ and $g: [0,1] \to [0,1]$ be two Riemann integrable functions. Assume that $|g(x) - g(y)| \geq \alpha |x-y|$ for any $x,y \in [0,1]$ and some fixed $\alpha \in (0,1)$. Show that $f \circ g$ is Riemann integrable.
Some thoughts have been bounding the intervals in which $f$ has a large oscillation by its integrability, and trying use the condition on $g$ to control the growth of these interval lengths. However, I am unsure how to apply the Riemann integrability of $g$.