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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 \\[\n  \\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). 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