Let $f$ be continuous on $[-1,2].$ for each of the increasing functions $g$ given below, explain why $f$ is Riemann–Stieltjes Integrable and compute$\int_{-1}^2 f\,dg$ in terms of Riemann integrals.
Computation:
$$g(x) = \begin{cases} -1 & \text{if }-1 \le x < 0, \\ 0 & \text{if } 0 \le x \le 1, \\ 1 & \text{if } 1 < x \le 2. \end{cases} $$
$f \in \mathbf{R}([a,b],g)$ as $f$ is continuous on $[-1,2],$ by Theorem 6.21.
$$\int_{-1}^2 f \, dg = \int_{-1}^0 f\,dg+ \int_0^1 f\,dg+ \int_1^2 f\,dg.$$
Since $g$ is differentiable on all three intervals $[-1,0],$ $[0,1],$ and $[1,2],$ we note that
$$ \int_{-1}^2 f\,dg= \int_{-1}^0 f(x)g'(x)\,dx + \int_0^1 f(x)g'(x) \, dx+ \int_1^2 f(x)g'(x) \, dx.$$
Thus, we've
$$\int_{-1}^2 f\,dg = \int_{-1}^0 f(x)g'(x) \, dx+ \int_0^1 f(x)g'(x)\,dx+ \int_1^2 f(x)g'(x)\,dx = 0$$