Approximation in $AC(\sigma)$

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.