ILL 1986 Problem 19

Let $f : [0, 1] \to [0, 1]$ satisfy $f(0) = 0, f(1) = 1$ and $$f(x + y) - f(x) = f(x) - f(x - y)$$ for all $x, y \geq 0$ with $x - y, x + y \in [0, 1].$ Prove that $f(x) = x$ for all $x \in [0, 1].$