  In this paper we define a notion of partial APNness and find various
characterizations and constructions of classes of functions satisfying this
condition. We connect this notion to the known conjecture that APN functions
modified at a point cannot remain APN. In the second part of the paper, we find
conditions for some transformations not to be partially APN, and in the
process, we find classes of functions that are never APN for infinitely many
extensions of the prime field $\F_2$, extending some earlier results of Leander
and Rodier.
