I presume that by "Lipschitz continuous" you mean what I would call just "Lipschitz": A function, f, is Lipschitz on a set A if and only if there exist a constant C such that, for all x, y in the set, |f(x)- f(y)|< C|x- y|.

The problem with that is that it requires a "set A". On what set do you want to prove this function is Lipschitz? The set of all real numbers?

It is relatively easy to prove, using the mean value theorem, that if a function is differentiable on a set then it is Lipschitz on that set. And this function is clearly differentiable for all x.