  We consider the behaviour of holomorphic functions on a bounded open subset
of the plane, satisfying a Lipschitz condition with exponent $\alpha$, with
$0<\alpha<1$, in the vicinity of an exceptional boundary point where all such
functions exhibit some kind of smoothness. Specifically, we consider the
relation between the abstract idea of a bounded point derivation on the algebra
of such functions and the classical complex derivative evaluated as a limit of
difference quotients. We show that whenever such a bounded point derivation
exists at a boundary point $b$, it may be evaluated by taking a limit of
classical difference quotients, for approach from a set having full area
density at $b$.

Complex Variables;Functional Analysis