Suppose $f(x)$ is a function that has derivative in the domain of our concern. We all know that if the derivative is bounded then $f$ is Lipschitz continuous. I was wondering if the above statement is an if-and-only-if statement since if the derivative is unbounded then surely we can find two points that fail the Lipschitz continuity property?
Many thanks in advance!
Yes it is.
Assume that $ f$ is differentiable and Lipschitz at $ A$, then there exists some $K>0$ such that for all $(x,x_0)$ in $A$, with $x\ne x_0$ ,
$$|\frac{f(x)-f(x_0)}{x-x_0}|\le K$$
then if we pass to the limit when $x\to x_0$,
$$|f'(x_0)|\le K$$
the derivative is necessarily bounded at $A$.