  Starting from the Colombeau's full generalized functions, the sharp
topologies and the notion of generalized points, we introduce a new kind
differential calculus (for functions between totally disconnected spaces). We
study generalized pointvalues, Colombeau's differential algebra, holomorphic
and analytic functions. We show that the Embedding Theorem and the Open Mapping
Theorem hold in this framework. Moreover, we study some applications in
differential equations.
