, 2014

Let $z > 2$. We consider the function $g: S \to \mathbb{R}$ with $g(x) = x^2 - z^2/x^2$. As $x \mapsto x^2$ and $x \mapsto -z^2/x^2$ are both increasing functions for $x > 0$, $g$ is also increasing and obviously continuous. As $g(1) = 1 - z^2 < 0$ and $g(x) = z^2 - 1 > 1$, there is an $x_0 \ge 1$ such that $g$ is a bijection from the interval $I = [x_0, z]$ to the interval $[1, z^2 - 1]$. For $x \in I$ and $y = z/x$, we have
$$x \ge 1, \quad y \ge 1, \quad 1 \le x^2 - y^2 = g(x) \le z^2 - 1,$$
so that the functional equation yields
$$f(z) = f(xy) = f(x^2 - y^2) = f(g(x))$$
for all $x \in I$. We conclude that $f$ is constant on the interval $[1, z^2 - 1]$.
As $z \ge 2$ was arbitrary, we conclude that $f$ is constant on all these intervals and therefore on $S$.
On the other hand, every constant function $f: S \to S$ is a solution to the functional equation.