I came across the following theorem the other day,
"If $f:[a,b]\to \mathbb{R}$ is monotonic increasing, then $f$ is differentiable a.e."
If the take the standard Cantor-Lebesgue function then I see it is not differentiable on the Cantor set, which is measure zero. Great.
Now construct the analogous function for a fat-Cantor set. It is monotonically increasing, but this time the set of discontinuities (i.e. the fat-Cantor set) has positive measure.
What's going on?
Thanks