Functional equations which are exorbitant (Q1193293)

The author calls a functional equation or a system of functional equations, containing the unknown functions \(f_ j:A_ j\to B_ j\) \((j=1,\ldots,m)\), exorbitant if the cardinality of the set of all solutions equals that of the set \(\{(g_ 1,\ldots,g_ m)\mid g_ j:A_ j\to B_ j\) \((j=1,\ldots,m)\}\) of all functions mapping \(A_ j\) into \(B_ j\) \((j=1,\ldots,m)\). He shows about several classes of functional equations that all equations in them are exorbitant, including the Cauchy equation \(f(x+y)=f(x)+f(y)\) and the Gołąb-Schnizel equation \(f(x+yf(x))=f(x)f(y)\). The last theorem in the paper states that a functional equation in one unknown real function, which is satisfied by all idempotent \((g(g(x))=g(x))\) additive functions, is exorbitant.