  For a nonempty compact subset $\sigma$ in the plane, the space $AC(\sigma)$
is the closure of the space of complex polynomials in two real variables under
a particular variation norm. In the classical setting, $AC[0,1]$ contains
several other useful dense subsets, such as continuous piecewise linear
functions, $C^1$ functions and Lipschitz functions. In this paper we examine
analogues of these results in this more general setting.
