  In this article we study the existence of pathwise Stieltjes integrals of the
form $\int f(X_t)\, dY_t$ for nonrandom, possibly discontinuous, evaluation
functions $f$ and H\"older continuous random processes $X$ and $Y$. We discuss
a notion of sufficient variability for the process $X$ which ensures that the
paths of the composite process $t \mapsto f(X_t)$ are almost surely regular
enough to be integrable. We show that the pathwise integral can be defined as a
limit of Riemann-Stieltjes sums for a large class of discontinuous evaluation
functions of locally finite variation, and provide new estimates on the
accuracy of numerical approximations of such integrals, together with a change
of variables formula for integrals of the form $\int f(X_t) \, dX_t$.
