  We introduce a new concept, the APN-defect, which can be thought of as
measuring the distance of a given function $G:\mathbb{F}_{2^n} \rightarrow
\mathbb{F}_{2^n}$ to the set of almost perfect nonlinear (APN) functions. This
concept is motivated by the detailed analysis of the differential behaviour of
non-APN functions (of low differential uniformity) $G$ using the so-called
difference squares. We describe the relations between the APN-defect and other
recent concepts of similar nature. Upper and lower bounds for the values of
APN-defect for several classes of functions of interest, including
Dembowski-Ostrom polynomials are given. Its exact values in some cases are also
calculated. The difference square corresponding to a modification of the
inverse function is determined, its APN-defect depending on $n$ is evaluated
and the implications are discussed.
  In the forthcoming second part of this work we further examine modifications
of the inverse function. We also study modifications of classes of functions of
low uniformity over infinitely many extensions of $\mathbb{F}_{2^n}$. We
present quantitative results on their differential behaviour, especially in
connection with their APN-defects.
