  In the theory of complex valued functions of a complex variable arguably the
first striking theorem is that pointwise differentiability implies $C^{\infty}$
regularity. As mentioned in Ahlfors's standard textbook there have been a
number of studies proving this theorem without use of complex integration but
at the cost of considerably more complexity. In this note we will use the
theory of non-absolutely convergent integrals to firstly give a very short
proof of this result without complex integration and secondly (in combination
with some elements of the theory of elliptic regularity) provide a far reaching
generalization.
