Lipschitz parameter of tanh function

Question

Problem

I am wondering if there is any global Lipschitz parameter for $\tanh(x)=\frac{1-\exp(-2x)}{1+\exp(2x)}$ function. More generally, is there a general way to prove a function is Lipschitz and find its Lipschitz parameter?

Answer 1

You have $$|\tanh'(x)| = |1- \tanh^2(x)| \leq 1$$ (because $-1 \leq \tanh(x) \leq 1$).

So, by the mean value theorem, $\tanh$ is $1$-Lipschitz.