Proving Lipschitz Condition in multi-variables function

Question

Say I have $f:\mathbb{R}^2 \to\mathbb{R}$ and I need to find its Lipschitz Constant in case the function indeed satisfies the Lipschitz condition.
My question is: Does proving $f$ satisfies the condition in its second variable enough? Or do I need to prove it for both of them?

Answer 1

A function $f:\>\Omega\to{\mathbb R}$ is Lipschitz continuous on its domain $\Omega\subset{\mathbb R}^n$ if there is a number $L\geq0$ such that $$|f(y)-f(x)|\leq L|y-x|\qquad\forall x, y\in\Omega\ .\tag{1}$$ This $L$ is by no means uniquely determined. It is sufficient to produce one $L$ such that $(1)$ is satisfied. Of course there is then an "optimal" $L$, which is the $\inf$ of all admissible $L$s, but finding this optimal $L$ is not required in general.

Lipschitz continuity involves the "full function" $f$ in all its variables. A simple sufficient condition is that the first partial derivatives should be continuous and bounded.

There is a certain area where Lipschitz continuity plays a decisive rôle, namely existence and uniqueness of solutions of IVPs $y'=f(x,y)$, $y(x_0)=y_0$. Here it is required that the right hand side $f(x,y)$ is continuous in both variables, and in addition Lipschitz continuous with respect to the second variable.