# 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