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
3
$\begingroup$
real-analysis
-
0This 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. – 2011-05-02
-
0This 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. – 2011-05-02
-
0What can you say about $|F(c+h) - F(c)|$ where $0 \lt |h| \lt \varepsilon$ with $\varepsilon$ small? – 2011-05-02
-
0Taken small enough, that would be pretty close to zero. – 2011-05-02
-
0So, that's continuity at $c$, isn't it? – 2011-05-02
-
0Thank you, I didn't see such a short argument. I wanted to throw every theorem at it from the R-S Integral chapter. – 2011-05-02
-
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