Let $f$ and $g$ be continuous and monotone increasing functions on $[a,b]$. Then the integration by parts formula for Lebesgue-Stieltjes integrals \begin{equation*} \int_a^b fdg+\int_a^b gdf =f(b)g(b)-f(a)g(a). \end{equation*}
I have proved that this is true, but can anyone come up with an example to show that this formula does not remain true without the continuity assumption. Any help would be appreciated.