Question 77
Let $S$ be a proper subset of $\mathbb{R}$ (i.e. $S \neq \mathbb{R}$) having at least two elements. Suppose there exists a function $f: \mathbb{R} \to \mathbb{R}$ satisfying the following conditions:
(i) $f(a + x + y) + f(f(a)) + f(x) + f(y) = x + y$; and
(ii) $f(axy) + f(a) + f(x)f(y) = xy$
for any real numbers $a \notin S$ and $x, y \in S$. Find all such function(s) $f$.
(i) $f(a + x + y) + f(f(a)) + f(x) + f(y) = x + y$; and
(ii) $f(axy) + f(a) + f(x)f(y) = xy$
for any real numbers $a \notin S$ and $x, y \in S$. Find all such function(s) $f$.