Removable singularities in C*-algebras of real rank zero

Let AA be a C*-algebra with identity and real rank zero. Suppose a complex-valued function is holomorphic and bounded on the intersection of the open unit ball of AA and the identity component of the set of invertible elements of AA. We give a short transparent proof that the function has a holomorphic extension to the entire open unit ball of AA. The author previously deduced this from a more general fact about Banach algebras.