The Cantor function — continuous, monotone, yet almost everywhere flat
6
1/3
Derivative = 0 almost everywhere, yet f(0)=0, f(1)=1
Flat steps cover all of [0,1]
Cursor x: —, f(x): —
What makes it strange: The Cantor function f is continuous and non-decreasing, f(0)=0, f(1)=1 — so it looks like a valid CDF. Yet f'(x)=0 Lebesgue-almost-everywhere (it's flat on every interval removed from the Cantor set, and the Cantor set has measure zero). The "climb" from 0 to 1 happens entirely on a set of measure zero. Varying α below 1/3 creates a fat-Cantor version where the removed intervals no longer cover everything, and the function gets a genuine positive-derivative component.