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Nice Functional Equations (ISI 2021)

Source: ISI 2021 P2

July 18, 2021
functional equationisiISI entrance

Problem Statement

Let f:Z→Zf : \mathbb{Z} \to \mathbb{Z} be a function satisfying f(0)≠0=f(1)f(0) \neq 0 = f(1). Assume also that ff satisfies equations (A) and (B) below. \begin{eqnarray*}f(xy) = f(x) + f(y) -f(x) f(y)\qquad\mathbf{(A)}\\ f(x-y) f(x) f(y) = f(0) f(x) f(y)\qquad\mathbf{(B)} \end{eqnarray*} for all integers x,yx,y.
(i) Determine explicitly the set {f(a) : a∈Z}\big\{f(a)~:~a\in\mathbb{Z}\big\}. (ii) Assuming that there is a non-zero integer aa such that f(a)≠0f(a) \neq 0, prove that the set {b : f(b)≠0}\big\{b~:~f(b) \neq 0\big\} is infinite.
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