Sorry for that regrettably late reply; an elementary example of function with this property is the real logarithm.
Neutrices were introduced, as far as I know, (I am not quite knowledgeable in the area) to help statisticians/theoretical physicists deal with asymptotic expressions, i.e. they generalize
O and
o notions. A neutrix is an abelian (additively written) group
N, consisting of functions
S→G, where
G is an abelian group, such that the only constant function in
N is the zero function. The functions in
N are called negligible.
As an example consider
S=Z,
G=R, and the negligible functions in
N being finite (linear) sums of
nxlnk−1n and
lnkn, where
x>0,
k - positive integer, and all functions which tend to zero as
n→∞.
This article is relevant.