Prove f is not Riemann-Stieltjes integrable

Question

f:[a,b]$\to \mathbb{R}$. If c$\in$(a,b) and f is not continuous at c,show that there exists an increasing function g:[a,b]$\to \mathbb{R}$ such that f is not Riemann-Stieltjes integrable on [a,b] with respect to g.

I cant figure out a way to solve this problem.Can anyone tell me how to approach this problem.

Answer 1

Hint:
For all we know, $f$ is continuous everywhere but at $c$ and if $\alpha(x)$ (where $\alpha(x)$ is the monotonically increasing function that we integrate $f$ with respect to) is continuous at every discontinuity of $f$ (for $f$ with finitely many discontinuities) then $f\in\mathcal{R}(\alpha)$. To avoid this, we must pick an $\alpha$ that is discontinuous at $c$. Then show that for some nice $\alpha(x)$, if $f\in\mathcal{R}(\alpha)$ then $f$ is continuous at c.

This $\alpha(x)$ should work well:

$\alpha(x)=\left\{ \begin{array}{lr} 0 & x<c\\ \frac{1}{2} & x=c\\ 1 & x>c \end{array} \right.$