I'd like to prove the following, which I'm reasonably sure is true:
Proposition. If $f : [a,b] \to \mathbb{R}$ is Riemann integrable and $g : \mathbb{R} \to \mathbb{R}$ is continuous and nondecreasing, then $g \circ f : [a,b] \to \mathbb{R}$ is Riemann integrable.
My idea is to use the Darboux definition of the integral: Fix $\epsilon > 0$ and let $h_1 \leq f \leq h_2$ be piecewise constant functions such that $\int_a^b h_2(x)\,dx - \int_a^b h_1(x)\,dx \leq \epsilon$. Then $g\circ h_1 \leq g \circ f \leq g \circ h_2$, where the outer functions are Riemann integrable (since they're piecewise continuous). The problem is bounding $\int_a^b g \circ h_2(x)\,dx - \int_a^b g \circ h_1(x)\,dx$ by a suitable function of $\epsilon$, since it's hard to tell how $g$ will distort the regions under these curves. How should I proceed? (I'd like to prove this straight from the definition of Darboux integrability, rather than from the fact that Riemann integrability is equivalent to continuity a.e.)
