For $f \in R([a,b], \alpha)$ define
$F(x) = \int_a^xfd\alpha, x\in [a,b]$
show that :
i) $\alpha$ is continuous at some $x\in [a,b]\Rightarrow F$ is continuous at $x$
ii)$\exists \alpha'(x)$ and $f$ is continuous at $x$ for some $x \in [a,b]$$\Rightarrow$ $\exists F'(x)$ and $F'(x)=f(x)\alpha'(x)$
How to prove above claim?
Intuitively i)states that only existence of continuous point of $\alpha$ could guarantee the continuity of $F$. However, I can't imagine how this could be possible.