Please, what is the difference between the following expressions "Lipschitz on any bounded set" and "Locally Lipschitz"?
The function $\sqrt x$ is locally Lipschitz on $(0,1)$ but is not Lipschitz on the bounded see $(0,1)$. This example is possible because $(0,1)$ is not compact.
As far as I can tell, the usual concept is Lipschitz on compact sets instead of on bounded sets. It is easy to prove that locally Lipschitz is equivalent to Lipschitz on compact sets.