This is one of those things where a surprising fact isn't astounding, but just makes me think the definitions are badly chosen. After all -- of course if you pick definitions in weird ways, you get weird results.
In this case, yes, it is true almost-everywhere, meaning that there the set of points where it is untrue has measure 0. 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. All this definition has done is crammed all of the increasing-ness into points that are vanishingly rare, so you can't find them. But they still exist, per _my_ definition of increasing!
I would instead choose to define "having a derivative != 0" in a different way, that captures the idea that, over any interval I choose, the function changes. Nevermind that the function becomes constant as I zoom in to a point. All that shows is that evaluating at points must not be a great way of modeling this.
Of course I'm not sure if that works exactly. But this is the reason I can't get enthusiastic about analysis: 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.
In this case, yes, it is true almost-everywhere, meaning that there the set of points where it is untrue has measure 0. 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. All this definition has done is crammed all of the increasing-ness into points that are vanishingly rare, so you can't find them. But they still exist, per _my_ definition of increasing!
I would instead choose to define "having a derivative != 0" in a different way, that captures the idea that, over any interval I choose, the function changes. Nevermind that the function becomes constant as I zoom in to a point. All that shows is that evaluating at points must not be a great way of modeling this.
Of course I'm not sure if that works exactly. But this is the reason I can't get enthusiastic about analysis: 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.