Lipschitz Condition
Let f be defined on a Domain or closed domain D of the XY plane.
f is said to satisfy Lipschitz condition (wrt y) in D, if
$$ \exists K(\forall (x,y_{1}),(x,y_{2})\in D(f(x,y_{1})-f(x,y_{2})\le K|y_{1}-y_{2}|)) $$
Sufficient condition #
By Mean Value Theorems of two variable functions, if $\frac{\partial f}{\partial y}$ is bounded by M,
$$ \begin{align*} \exists E\in[y_{1},y_{2}]: f(x,y_{1})-f(x,y_{2})=(y_{1}-y_{2}) \frac{\partial f(x,E)}{\partial y}\\ |f(x,y_{1})-f(x,y_{2})|<|y_{1}-y_{2}|M \end{align*} $$
Therefore, sufficient condition for Lipschitz condition is for $\frac{\partial f}{\partial y}$ to be bounded where lipscitz constant =
$K=M=lub_{(x,y)\in D}(\frac{\partial f}{\partial y})$