The article links to the Wikipedia page, which gets it right.
"An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has a bounded first derivative"
Lipschitz continuity for differentiable functions is just having a bounded derivative. The Lipschitz property suddenly becomes uninteresting as it just falls out of the assumptions, the interesting fact which allows you to use non-differentiable functions is that not assuming differentiability, but assuming Lipschitz continuity, is enough.
>Using that bound is bread and butter in convex analysis of differentiable convex functions.
It also is the bread and butter in Analysis of PDEs, but it is the bread and butter because Lipschitz continuity is a weaker property than differentiability. In the context of the article you want to talk about non-differnetiable functions, e.g. max, which you couldn't if you assumed differentiability.
The reason this is important because choosing Lipschitz over differentiable is what makes all this work.
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.
The article links to the Wikipedia page, which gets it right.
"An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has a bounded first derivative"
Lipschitz continuity for differentiable functions is just having a bounded derivative. The Lipschitz property suddenly becomes uninteresting as it just falls out of the assumptions, the interesting fact which allows you to use non-differentiable functions is that not assuming differentiability, but assuming Lipschitz continuity, is enough.
>Using that bound is bread and butter in convex analysis of differentiable convex functions.
It also is the bread and butter in Analysis of PDEs, but it is the bread and butter because Lipschitz continuity is a weaker property than differentiability. In the context of the article you want to talk about non-differnetiable functions, e.g. max, which you couldn't if you assumed differentiability.
The reason this is important because choosing Lipschitz over differentiable is what makes all this work.