It's much simpler than it looks :-)
The curve that they are talking about is all corners - it has no smooth sections. There are no breaks in the curve, so you can call it "continuous", but each and every point of it is a jagged corner.
Typically by "differentiating" a thing they mean putting a wooden ruler between two points very near each other and recording which direction the ruler points at.
But as you can see here, if the curve is all corners you can't put a ruler anywhere because it lands on a corner, whatever you do :-) And since your ruler rocks every which way on that corner, the curve is not "differentiable" - there is no single direction the ruler would point to. That's all to it.
LOL... The last time I used a ruler.. Oh, nevermind..
From your post, I get the impression that the phenomena being discussed are simply artifacts of the explanatory maths.
Schrodinger never owned a damn cat, neither!
Thanks, Greysard.
There are 2 kinds of people in the world. Those who understand calculus and those who don't.
Those who don't understand calculus own 99% of the guns and know how to use them. The reason that you guys can commiserate about bizarre theories like fractals is that the rest of us appreciate smart guys and know that your type developed atomic bombs, computers and DDT.
Don't push it. :)