  Given a collection A of holomorphic functions, we consider how to describe
all the holomorphic functions locally definable from A. The notion of local
definability of holomorphic functions was introduced by Wilkie, who gave a
complete description of all functions locally definable from A in the
neighbourhood of a generic point. We prove that this description is no longer
complete in the neighbourhood of non-generic points. More precisely, we produce
three examples of holomorphic functions which suggest that at least three new
operations need to be added to Wilkie's description in order to capture local
definability in its entirety. The constructions illustrate the interaction
between resolution of singularities and definability in the o-minimal setting.
