Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a monotone increasing function. We know from a general fact that the set of discontinuities of $f$ is countable. Denote the set of discontinuities by $D$.
I want to construct a continuous function $g$ such that there exists a function $h$ defined on $D$ with $f(b)-f(a)=g(b)-g(a)+\sum_{x\in (a,b)\cap D}h(x)$.
The problem is that there can be a countable number of discontinuities, and it is not necessary that we can order them as $\cdots<x_{-2}<x_{-1}<x_0<x_1<x_2<\cdots$. How can we construct this function $g$?
