TITLE: How to compute Lipschitz Constant for multivariate function $f(x,y)=1-xy$?
QUESTION [7 upvotes]: How to compute Lipschitz Constant for multivariate function $f(x,y)=1-xy$? I know the definition for one variable? What is its definition for multivariate functions?

REPLY [12 votes]: As $f$ is a quadratic polynomial in $x$ and $y$ you cannot expect that there is a globally valid Lipschitz constant for $f$, because such a constant would allow only a linear increase of $f$ on a large scale.
For a formal proof consider the point ${\bf z}_0:=(x_0,0)$ and the "nearby" point ${\bf z}:=(x_0,h)$, $\>h>0$. Then $|{\bf z}-{\bf z}_0|=h$ and
$$|f({\bf z})-f({\bf z_0})|=|(1-x_0h)-1|=|x_0|\>h=|x_0|\>|{\bf z}-{\bf z}_0|\ .$$
Since $|x_0|$ can be arbitrarily large it follows that there is no universal $C>0$ such that
$$|f({\bf z})-f({\bf z_0})|\leq C\>|{\bf z}-{\bf z}_0|$$
for all ${\bf z}$, ${\bf z}_0\in{\mathbb R}^2$.