VMO 1970 Problem 3

The function $f(x, y)$ is defined for all real numbers $x, y$ . It satisfies $f(x,0) = ax$ (where $a$ is a non-zero constant) and if $(c, d)$ and $(h, k)$ are distinct points such that $f(c, d) = f(h, k)$ , then $f(x, y)$ is constant on the line through $(c, d)$ and $(h, k)$ . Show that for any real $b$ , the set of points such that $f(x, y) = b$ is a straight line and that all such lines are parallel. Show that $f(x, y) = ax + by$ , for some constant $b$ .