Given $\alpha$ on $[a,b]$ and $f \in R(\alpha)$. For some $x \in [a,b]$, define $F(x) = \int_{a}^{x}f\,d\alpha$. Let $c \in [a,b]$. Prove that if $\alpha$ is continuous at $c$, then $F$ is continuous at $c$.
Continuity and Riemann-Stieltjes Integrals
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real-analysis
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0Well, you still need to prove this in detail, but there is not much more to it. Of course you need to use that $\alpha$ is continuous at $c$. – 2011-05-02