[Math] Determining the smallest possible Lipschitz constant

Question

For example if I have the function $\arctan(2x)$ then I can prove that the Lipschitz constant is 2. How can I prove that it won't work for constant less than 2?

Answer 1

Let $f(x)=\arctan(2\,x)$. Then $|f'(x)|\le2$,and that is how you know that $2$ is a Lipschitz constant for $f$. Since $f'(0)=2$, no smaller constant will do. Given $0<L<2$, from $\lim_{x\to0}f(x)/x=2$ we deduce that there is $\delta>0$ such the $|f(x)/x|\ge L$ if $|x|\le\delta$. Then $$ |x|\le\delta\implies |f(x)-f(0)|\ge L\,|x-0|. $$