Do there exist $f$ and $g$ continuous functions on $[1,2]$ such that $\int f dg$ does not exist ?
I tried with
$f(x)=1$
$g(x) = \begin{cases} (x-1)\sin\left(\frac{1}{x-1}\right) & \text{if } x \neq 1 \\ 0 & \text{if } x = 1 \end{cases}$
I chose $g$ which is not a bounded variation function.
Is it good example?
Another example is :
$f(x)= \begin{cases} (x-1)\sin\left(\frac{1}{x-1}\right) & \text{if } x \neq 1 \\ 0 & \text{if } x = 1 \end{cases}$
$g(x) = \begin{cases} (x-1)\sin\left(\frac{1}{x-1}\right) & \text{if } x \neq 1 \\ 0 & \text{if } x = 1 \end{cases}$
but how can I prove it ?
Riemann-Stieltjes integral is really hard, so proving existence or non existence is hard.