# Show f is Riemann Stieltjes Integrable

Let $f:[-1,1] \rightarrow \mathbb{R}$ be bounded and $g:[-1,1] \rightarrow \mathbb{R}$ be defined as $$g(x) = \left\{\begin{array}{ll} 0 & : x \in [-1,0]\\ 1 & : x \in (0,1] \end{array} \right.$$ Show that $f$ is Riemann Stieltjes integrable with respect to $g$ on $[-1,1]$ if and only if $$\lim_{x\to 0+}f(x) =f(0)$$

This is in mathematical analysis: Riemann- stieltjes integrals. I know If f is Riemann Stieltjes integrable then the upper integral = the lower integral, where the upper integral = inf U(f,g) and lower integral = sup L(f,g) Would anyone help on how to prove this on both ways around? Thank you

• Yeah, showed it! And what's your question? – Gono Mar 23 '17 at 14:59
• Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. Also, many find the use of imperative ("Prove", "Solve", etc.) to be rude when asking for help; please consider rewriting your post. – Clement C. Mar 23 '17 at 15:09
• @Sea Clearly, given the current edit you made, you didn't quite understand the point of my comment. it is helpful if you say in what context you encountered the problem, and what your thoughts on it are – Clement C. Mar 23 '17 at 15:21