Let $f$, $g$: $[a, b] \to \mathbb{R}$, $\mathrm{Var}(p, f) < \infty$, $\mathrm{Var}(q, g) < \infty$, $\frac{1}{p} + \frac{1}{q} = 1$. Why then $\int_{a}^{b} fdg$ maybe it is not defined? There were ideas to take partial sums of the Weierstrass function: $f(x) = \sum\limits_{n = 1}^{N} 2^{-\frac{n}{2}}\cos(2^{n}x)$ and $g(x) = \sum\limits_{n = 1}^{N} 2^{-\frac{n}{2}}\sin(2^{n}x)$. They have a limited 2-variation, only now I can't understand how to construct the necessary partition so that the existence of the Riemann-Stieltjes integral is denied. If there are other examples, they are also welcome. Or can you give some literature where there is this example.
Thanks for your help!
