Find $f \in B[0,2] $ and $g$ Riemann-Stieltjes integrable on $[0,1]$ and $[1,2]$ but not in $[0,2]$

Question

I'm stuck at one exercise:

Find $f \in BV[0,2]$ and $g$ such that $g$ is Riemann-Stieltjes integrable in $[0,1]$ and $[1,2]$, that is, $$\int_{0}^1 \ g \ df$$ and $$\int_{1}^2 g \ df$$ exist, but $g$ is not RS with respect to $f$ in $[0,2]$.

I have an idea. I think $f$ must have some "singularity" in $x=1$ (left and right limits of $f$ are not equal, or someting like that), but I can't find such functions.

Thank you a lot.

Answer 1

Let $f(x)=0$ $ (0\le x\le 1)$, $ f(x)=1$ $ (1<x\le 2)$ and $g(x)=0$ $ (0\le x<1)$, $ g(x)=1$ $ (1\le x\le 2)$.
Then $\int_{0}^1 \ g \ df=0$ and $\int_{1}^2 \ g \ df=1$.
But $\int_{0}^2 \ g \ df$ doesn't exist.
Consider a partition $P=\{ 0 = x_0 < x_1 < \cdots < x_n = 2\}$ with any $x_i\ne 1$.
Let $x_k<1<x_{k+1}$.
If $c_k<1$ , then $\sum_{i=0}^{n-1} g(c_i)(f(x_{i+1})-f(x_i))=0$. If $c_k>1$ , then $\sum_{i=0}^{n-1} g(c_i)(f(x_{i+1})-f(x_i))=1$.
Thus $\lim_{n \to \infty} \sum_{i=0}^{n-1} g(c_i)(f(x_{i+1}) -f(x_i))$ doesn't exist.