$f,g$ are real functions.
$f(x_1)=g(x_2)$ for any $x=(x_1,x_2)$
$x\in\mathbb R^2$.
Prove that $g,f$ are constants.
My tries:
It seems really easy.
Just simply take $y=(x_1,y_2)$. We have, for any $x_2,y_2$:
$$g(x_2)=f(x_1)=g(y_2).$$
That is $g$ is constant. Similarly $f$ is constant.
I am not sure my logics is correct?