Let $f : [a, b] → R$ be a Riemann integrable function and let $F$ be a continuously differentiable function on $R$. Prove that $g(x) = F(f(x))$ is Riemann integrable on $[a, b]$.
I was given the following solution:
Consider the difference between upper and lower Riemann sum under a partition $P$ of $[a, b]$. On the $i$-th interval, $\overline{S_i}(F(f(x)),P)- \underline{S_i}(F(f(x)),P)=[F(f(x_i))-F(f(x_{i-1}))]\Delta x_i$ ... and the proof continues.
I stopped reading here and got confused. How does this equality hold? Is this a mistake? Isn't the author assuming that $F(f(x))$ is increasing, when that assumption was not given in the problem statement?
I agree with you: this statement depends on $F \circ f$ being a weakly increasing function, which is not an assumption in the claim you stated.
My best guess is that the solution is wrong. My second-best guess is that you've accidentally looked up a solution to a different exercise from the one you intended - like maybe a different edition of the book has a different exercise #3 that does involve increasing functions, or something like that.