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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 $c$.

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This looks a lot like homework. What have you tried to do and where did you get stuck? Let me add that some people here are allergic to questions asked in the imperative since they deem this impolite. –  t.b. May 2 '11 at 6:02
    
This is an old test question. I had some crazy answer that attempted to invoke closed invervals implying differentiability. I'm not sure how wrong I was though since the question got thrown out. –  emka May 2 '11 at 6:07
    
What can you say about $|F(c+h) - F(c)|$ where $0 \lt |h| \lt \varepsilon$ with $\varepsilon$ small? –  t.b. May 2 '11 at 6:10
    
Taken small enough, that would be pretty close to zero. –  emka May 2 '11 at 6:11
    
So, that's continuity at $c$, isn't it? –  t.b. May 2 '11 at 6:11

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