In chapter 16 of Mathematical Analysis Apostol defines contour integrals in terms of Riemann-Stieltjes integrals, specifically:
$$\int_\gamma f = \int_a^b f[\gamma(t)]\,d\gamma(t)$$ whenever the Riemann-Stieltjes integral on the right exists. My question would be "Find necessary and sufficient conditions on a contour $\gamma$ to ensure that the Riemann-Stieltjes integral defined above (and thus the contour integral) exists for all continuous complex functions $f$."
It is well-known that requiring $\gamma$ to be rectifiable is a sufficient condition for the contour integral to exist. But is it necessary? Is it the case that for every non-rectifiable curve $\gamma$ we can find a continuous function $f$ for which $\int_\gamma f$ (as defined above) does not exist? If it is not a necessary condition, what is?