Is this a Lipschitz continuous function?

Question

Is $$\frac{xy}{1 + x^2 + y^2}$$ Lipschitz continuous on $x^2 + y^2 \le 4$?

I've tried using Cauchy-Schwarz intequality but got nothing. I also tried to find out whether $xy$ is Lipschitz but failed as well.

Any hints on this problem?

EDIT: I need to exhibit the Lipschitz constant if there is one.

Answer 1

The function $f(x,y)=\frac {xy}{1+x^2+y^2}$ is $C^1$ because the partial derivatives are continuous.Thus it's Lipschitz con.