Is $f \left(x\right)=\left( x_1 + \frac{x_2 x_3}{x_1^2 + x_2^2}, x_2 - \frac{x_1 x_3}{x_1^2 +x_2^2}, -x_3 \right)'$ is a Lipschitz function?

Question

Given

$$f \left(x_1,x_2,x_3\right)=\left( x_1 + \dfrac{x_2 x_3}{x_1^2 + x_2^2}, x_2 - \dfrac{x_1 x_3}{x_1^2 +x_2^2}, -x_3 \right) \prime$$

Does anybody know the best method to prove that this function is a Lipschitz function?