  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 obtain a result which applies, for example, when the
open set admits an interior cone at the special boundary point.
