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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 $P_n=(x_0=a<x_1<...<x_n=b)$ 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?

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    $\begingroup$ 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. $\endgroup$ Aug 25, 2020 at 12:17
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    $\begingroup$ 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. $\endgroup$
    – zkutch
    Aug 25, 2020 at 12:43

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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 $x_0=a<x_1<...<x_n=b$ 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 $x=a$.

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