[all-fakes]

> And yet intuitively the function is increasing and so we ought to define "having a derivative" in such a way that shows that it is increasing.

The Cantor function is not differentiable at the points of "increase", so there is no problem with it growing at those points. I'm not sure what your complaint is.

> it seems like a bunch of silly definitions with silly consequences, with little relation to the actual mathematics of our universe that I care about.

That's honestly really silly. A lot of analysis is extremely practical, and is fundamental to the study of this universe. Yes, the language allows you to express non-physical situations, but you know you can just ignore those situations, right?