Question 3
To every point $(x,y)$ in the coordinate plane is assigned a real number $f(x,y)$. Suppose that:
(a) $f(x,0)=ax$, where $a$ is a nonzero constant;
(b) if two distinct points $(x_1,y_1),(x_2,y_2)$ have the same value of $f$, then $f$ is constant on the whole line through them.
Prove that:
(a) for every real $\alpha$, the set where $f(x,y)=\alpha$ is a line $D_\alpha$, and all these lines are parallel;
(b) $f(x,y)=ax+by$ for all $x,y$, where $b$ is a constant.
(a) $f(x,0)=ax$, where $a$ is a nonzero constant;
(b) if two distinct points $(x_1,y_1),(x_2,y_2)$ have the same value of $f$, then $f$ is constant on the whole line through them.
Prove that:
(a) for every real $\alpha$, the set where $f(x,y)=\alpha$ is a line $D_\alpha$, and all these lines are parallel;
(b) $f(x,y)=ax+by$ for all $x,y$, where $b$ is a constant.