How can I calculate a Lipschitz constant for a 2-dimensional real-valued $C^{\infty}$ function with bounded derivatives?
[Math] How to calculate Lipschitz constant
calculuslipschitz-functionsreal-analysis
calculuslipschitz-functionsreal-analysis
How can I calculate a Lipschitz constant for a 2-dimensional real-valued $C^{\infty}$ function with bounded derivatives?
Best Answer
By the mean value theorem $$ f(x)-f(y)=\nabla f(\xi)\cdot(x-y) $$ for some $\xi$ in the segment joining $x$ and $y$. Since $f$ has bounded partial derivatives, there is an $M>0$ such that $|\nabla(x)|\le M$ for all $x\in\mathbb{R}^2$. Then $$ |f(x)-f(y)|\le M\,|x-y|. $$ ($|z|$ is the Euclidean norm in $\mathbb{R}^2$ of $z=(z_1,z_2)$.)