My point is that the article is using the Lipschitz property to get its results. This makes it unnecessary and even wrong to introduce Lipschitz continuity only for functions with a derivative. Especially since the article actually uses functions which do not have a derivative.
For what it's worth, what you originally said was: "This is total nonsense." The points you're making are valid, but it isn't "total nonsense". Something not being exactly factually correct doesn't mean that it's "total nonsense". Different publications adhere to different levels of rigor. Just because it doesn't meet your own personal standard doesn't make it nonsense for the target audience of the article.
It gives the special property that the derivative is bounded, a property that's not true for arbitrary differentiable functions.