  We prove that functions $f:\f{2^m} \to \f{2^m}$ of the form
$f(x)=x^{-1}+g(x)$ where $g$ is any non-affine polynomial are APN on at most a
finite number of fields $\f{2^m}$. Furthermore we prove that when the degree of
$g$ is less then 7 such functions are APN only if $m \le 3$ where these
functions are equivalent to $x^3$.
