  We define the algebra of Colombeau generalized functions on a subset A of the
space of d-dimensional generalized points. If the domain A is open, such
generalized functions can be identified with pointwise maps from A into the
ring of Colombeau generalized numbers. We study analyticity in these algebras,
if the domain is an open subset of the complex generalized points. In
particular, if the domain is an open ball for the sharp norm, we characterize
analyticity and give a unicity theorem involving the values at generalized
points.
