Lipchitz continuity for two variables function

Question

I have a function $f:\mathbb{R}^{d} \times \mathbb{R}^{p} \longrightarrow \mathbb{R}^{2}$.

I have demonstrated that $f$ is Lipschitz continuous for each variable separately.

My question: is it enough to conclude that $f$ is Lipschitz continuous ?

Answer 1

Yes, it is enough. Suppose that the Lipschitz constants for each variable are $L_1$ and $L_2$. Then for any two points $(p_1,q_1)$ and $(p_2,q_2)$, we have

$$d(f(p_1,q_1)),f(p_2,q_2))\leq d(f(p_1,q_1),f(p_1,q_2))+d(f(p_1,q_2),f(p_2,q_2))\leq L_2d(q_1,q_2)+L_1d(p_1,p_2)\leq (L_1+L_2)d((p_1,q_1)),(p_2,q_2)),$$

so the Lipschitz constant $L_1+L_2$ works.