  On a reflexive Banach space $X$, if an operator $T$ admits a functional
calculus for the absolutely continuous functions on its spectrum $\sigma(T)
\subseteq \mathbb{R}$, then this functional calculus can always be extended to
include all the functions of bounded variation. This need no longer be true on
nonreflexive spaces. In this paper, it is shown that on most classical
separable nonreflexive spaces, one can construct an example where such an
extension is impossible. Sufficient conditions are also given which ensure that
an extension of an $\AC$ functional calculus is possible for operators acting
on families of interpolation spaces such as the $L^p$ spaces.
