ISL 1989 Problem 52

Let $f$ be a function from the real numbers to the real numbers such that $f(1) = 1, f(a+b) = f(a)+f(b)$ for all $a, b,$ and $f(x)f \left( \frac{1}{x} \right) = 1$ for all $x \neq 0.$ Prove that $f(x) = x$ for all real numbers $x.$