Let $\alpha:[a,b]\rightarrow \mathbb{R}$ be a monotonically increasing function.
Let $f\in \mathscr{R}(\alpha)$ on $[a,b]$.
And here's what i have proved a while ago;
"If $\alpha$ is continuous at $a$, then $\lim_{x\to a}\int_{x}^{b} f d\alpha = \int_{a}^{b} f d\alpha$" (Existence of the limit is the part of the conclusion)
However, now i just realized 'continuity of $\alpha$' shouldn't be essential, but i cannot prove this. How do i prove this?