Is $f(x,y) = (x^2-y^2,xy)$ lipschitz on $\mathbb{R}^2$?

Question

How can I show that the $f$ is lipschitz? I try to calculation such that

$|f(x_0, y_0) - f(x_1,y_1)|^2 = ((x_0^2 -y_0^2)-(x_1^2 -y_1^2))^2 +(x_0 y_0 -x_1y_1)^2$ and $|(x_0-x_1,y_0-y_1)|^2 = (x_0-x_1)^2+(y_0-y_1)^2 $

But I don't know what can I do further.

Answer 1

If you restrict $f$ to the real line you get $f(x,0) = (x^2,0)$ which isn't Lipschitz. Thus it isn't Lipschitz on the whole space, either.