Bounded point derivations on certain function algebras

Let X be a compact nowhere dense subset of the complex plane ℂ, let C(X) be the linear space of all continuous functions on X endowed with the uniform norm, and let dA denote two-dimensional Lebesgue (or area) measure in ℂ. Denote by R(X) the closure in C(X) of the set of all rational functions having no poles on X. It is well known that if X is sufficiently massive, then the functions in R(X) can inherit many of the properties usually associated with the analytic functions, such as unlimited degrees of differentiability and even the uniqueness property itself. Here we shall examine the extent to which some of those properties are inherited by the larger algebra H^∞(X), which by definition is the weak-* closure of R(X) in L^∞(X) = L^∞(X, dA).