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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?

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According to Wheeden and Zygmund's text, if $f$ and $\alpha$ share a discontinuity, then the Riemann-Stieltjes integral does not exist. This means that we must assume $\alpha$ continuous at $a$, otherwise $f$ and $\alpha$ can share a discontinuity there.

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    Glad it makes sense!2012-12-21