Assume that $f$ is a bounded increasing function over an interval $I$ (possibly an unbounded interval) Then we know that $f$ is almost everywhere differentiable. How would one prove that :
$$ \sup_I f - \inf_I f = \int_I f' $$
2026-03-27 00:09:29.1774570169
Fundamental theorem of calculus for a bounded increasing function
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You wouldn't (i.e. this isn't true). Take for example the Cantor function which has almost everywhere derivative $0$ but clearly changes so that the difference between the sup and inf is non-zero.