Given $g:[a,b] \to [c,d]$ and $f:[c,d] \to \mathbb R$ as two functions such that $g$ is a polynomial and $f$ is monotonic, Prove that $f\circ g$ is Riemann integrable on $[a,b]$.
I found a similar theorem about Lebesgue integration, but it doesn't work for Riemann integration. I think the proof should be related to the powers of $f$. I don't know how to write a formal and neat proof...
If $g$ is constant the result is trivial, so assume $g$ is not constant. Recall that a function is Riemann integrable on $[a,b]$ iff it is bounded and its set of discontinuities has Lebesgue measure zero. Since $f$ is monotonic, it has at most countably many discontinuities. The only possibly points at which $f \circ g$ can fail to be differentiable are $x$'s such that $g(x)$ is a discontinuity point of $f$. For each such discontinuity point $y$, the equation $g(x) = y$ has at most finitely many solutions because $g$ is a non-constant polynomial. Thus the set of discontinuities of $f \circ g$ is countable and hence has Lebesgue measure zero. Since $f \circ g$ is bounded by $f(c)$ and $f(d)$, we find that $f \circ g$ is Riemann integrable.