TITLE: Why is the Cantor function not absolutely continuous?
QUESTION [33 upvotes]: Is there an easy way to see that the Cantor function is not absolutely continuous that directly uses the definition of absolutely continuous?

REPLY [2 votes]: A function is absolutely continuous if and only if it satisfies the (Lebesgue) fundamental theorem of calculus. Since $f'=0$ a.e., $f$ satisfies the fundamental theorem calculus if and only if 
$$f(1)-f(0) = \int_{0}^{1} f' = 0 $$
which we know is false.