In this case $g$ is monotone increasing on the finite interval $[a,b]$. Does this mean that $f$ is continuous in the normal sense of continuity (ie. $|f(x)-f(y)| < \epsilon$ when $|x-y|<\delta$) except on a set of measure $0$ where the measure here is the measure induced by $g$?
For clarification here is he whole question I am asked to answer.
Let $g$ be a monotone function on a finite closed interval $[a,b]$. Show that a bounded function $f$ defined on $[a,b]$ is Riemann-Stieltjes integrable with respect to $g$ iff $f$ is continuous almost everywhere with respect to the measure induced by $g$.
Im guessing the measure induced by $g$ is the Borel measure, $\mu_g$, with the property $\mu_g(a,b) = g(b)-g(a)$ since $g$ is monotone.