# How to calculate Lipschitz constant?

How can I calculate a Lipschitz constant for a 2-dimensional real-valued $C^{\infty}$ function with bounded derivatives?

• use Taylor's theorem – Quickbeam2k1 Mar 14 '15 at 12:35

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 $z=(z_1,z_2)$.)