The Cantor staircase is an example of a continuous function $f$ such that $f'=0$ almost everywhere, and yet $f$ is nonconstant. It is differentiable precisely at the points that aren't in the Cantor set.
Now, repeat the construction with a fat Cantor set (such as the Smith–Volterra–Cantor set, shown here).
Where is this fat Cantor staircase differentiable? Naively, I would've expected it to be differentiable on the complement of the Cantor set again, just like the original staircase. But this contradicts the theorem that says monotone functions are differentiable almost everywhere. So the fat Cantor staircase must be differentiable on some points in the Cantor set! Where, precisely, in the Cantor set does this happen? What is the derivative there? And what does the integral of this function's derivative look like?
(Conjecture: it is differentiable on the "pseudo-interior" - the set of points in the fat Cantor set without successor or predecessor, not counting 0 and 1.)
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