  This paper deals with stochastic integrals of form $\int_0^T f(X_u)d Y_u$ in
a case where the function $f$ has discontinuities, and hence the process $f(X)$
is usually of unbounded $p$-variation for every $p\geq 1$. Consequently,
integration theory introduced by Young or rough path theory introduced by Lyons
cannot be applied directly. In this paper we prove the existence of such
integrals in a pathwise sense provided that $X$ and $Y$ have suitably regular
paths together with some minor additional assumptions. In many cases of
interest, our results extend the celebrated results by Young.
