# Riemann-Stieltjes integral question

Let $$f.g:[a,b] \to \mathbb{R}$$, $$g(x)=\begin{cases} 0, & x=a \\ 1, & x\in (a,b] \end{cases}$$. Prove that $$f$$ is Riemann-Stieltjes integrable with respect to $$g$$ if and only if $$f$$ is continuous at $$a$$ and that in this case the integral is equal to $$f(a)$$.
I have just started learning the Riemann-Stieltjes integral, so I would like to do this by using the definition (by the way, while I was reading https://en.wikipedia.org/wiki/Riemann%E2%80%93Stieltjes_integral I came across that "generealized Riemann-Stieltjes" integral and, at least to me, that definition with refinements of partitions is equivalent to the first one where the norm approaches $$0$$. Am I right?). If I pick a sequence of partitions $$P_n=(x_0=a and the arbitraryintermediary points $$(c_1, c_2, ..., c_n)$$, then $$\lim\limits_{n\to \infty}S(P_n, f, g)=\lim\limits_{n\to \infty}f(c_1)$$. Now, I can see intuitively why the conclusion of the problem should follow, but how should I go about proving this in a formal way?

• Write down the upper and lower sums corresponding to a certain partition, and see what their difference is. This difference goes to zero if and only if the function is R-S integrable, so you will get a rigorous proof that way. You've pretty much done that work above. Aug 25 '20 at 12:17
• Your function $g$ is Heaviside step function with shifted $0 \to a$. It's Riemann-Stieltjes integral is considered, for example, Vladimir A. Zorich - Mathematical Analysis volume I- (2016) page 586. Aug 25 '20 at 12:43

First of all, let me make remark, that we are speaking about $$f$$ continuity in $$x=a$$ from right.
As it comes from definition, function $$g$$ have discontinuity in point $$x=a$$ and have jump discontinuity with jump $$1$$.
Assume, that $$f$$ is continuous at point $$x=a$$ and let's consider partition $$x_0=a and Stieltjes sum $$\sum\limits_{i=0}^{n-1}f(\xi_i)[ g(x_{i+1}-g(x_i)]=f(\xi_0)g(x_0)=f(\xi_0)$$ Now, when maximum size of a partition element goes to $$0$$, $$\xi_0$$ tends to $$x=a$$ from right and we have $$(S)\int\limits_{a}^{b}f(x)\,dg(x)=f(a)$$
Now assume, that exists Stieltjes integral and equals $$f(a)$$. This means, that for any choice of $$\xi$$ values Stieltjes sum limit exists. Taking in place of $$\xi$$ any sequence which converges from right to $$a$$ we obtain continuity of $$f$$ from right in $$x=a$$.