  The main purpose of this paper is to give characterization theorems on
derivations as well as on linear functions. Among others the following problem
will be investigated: Let $n\in\mathbb{Z}$, $f, g\colon\mathbb{R}\to\mathbb{R}$
be additive functions, $<{array}{cc} a&b c&d
{array}>\in\mathbf{GL}_{2}(\mathbb{Q})$ be arbitrarily fixed, and let us assume
that the mapping \[
  \phi(x)=g<\frac{ax^{n}+b}{cx^{n}+d}>-\frac{x^{n-1}f(x)}{(cx^{n}+d)^{2}} \quad
<x\in\mathbb{R}, cx^{n}+d\neq 0> \] satisfies some regularity on its domain
(e.g. (locally) boundedness, continuity, measurability). Is it true that in
this case the above functions can be represented as a sum of a derivation and a
linear function? Analogous statements ensuring linearity will also be
presented.
