I am finding an alternative proof to the following:
Suppose $f\in\mathcal{R}([a,b])$ and $g\in\mathcal{C}^{0,\alpha}(f([a,b]))$, then $g\circ f\in\mathcal{R}([a,b])$.
I want to point out that, I know how to prove it with Lebesgue criterion but I am avoiding it. I also know how to prove the more generalized version, with $g\in\mathcal{C}^0(f([a,b]))$, which splits the partition into two classes depending on $ \omega(f,S\in\mathcal{P}) < δ$ or not.
I have already proved the Lipschitz case, that is $g\in\mathcal{C}^{0,1}(f([a,b]))$ without splitting the partitions, so I suspect that we can generalize it a bit. Here is my attempt.
Let $\varepsilon >0$, since $g\in\mathcal{C}^{0,\alpha}(f([a,b]))$, there exist $\alpha >0$ and $K\geq 0$ such that $$\forall y,v\in f([a,b]), \qquad \left\lvert\vphantom{\frac{}{}} g(y) - g(v)\right\rvert \leq K\lvert y - v\rvert^\alpha.$$
If $K = 0$ then it is trivial, so suppose $K > 0$. Since $f\in\mathcal{R}([a,b])$, there exists a partition $\mathcal{P}$ of $[a,b]$ such that $$\sum_{S\in\mathcal{P}}\omega(f,S)\ell(S) < \left(\frac{\varepsilon}{K}\right)^{1/\alpha} .$$
Now $$\begin{aligned}\sum_{S\in\mathcal{P}}\omega(g\circ f,S)\ell(S) &= \sum_{S\in\mathcal{P}}\sup_{x,u\in S}\left\lvert\vphantom{\frac{}{}} g(f(x)) - g(f(u))\right\rvert\ell(S) \\ &\leq \sum_{S\in\mathcal{P}}\sup_{x,u\in S}K\left\lvert\vphantom{\frac{}{}} f(x) - f(u)\right\rvert^\alpha\ell(S) \\ &= K\sum_{S\in\mathcal{P}}\omega(f,S)^\alpha\ell(S) \\ &\overset{?}{\leq} K\left(\sum_{S\in\mathcal{P}}\omega(f,S)\ell(S)\right)^\alpha < K \left(\frac{\varepsilon}{K}\right) = \varepsilon.\end{aligned}$$
The problem is, I think the $?$ part is not true in general. Can someone shed some light on me?
