# Positive derivative almost everywhere?

is it true that if an absolutely continuous function defined on a closed interval in $\mathbb{R}$ has a non-negative derivative almost everywhere (non negative wherever the derivative is defined), then such function is monotone? (increasing, but not strictly)

• Yes. Because $f(y)-f(x) = \int_x^y f'(t) dt$. – copper.hat Nov 6 '14 at 6:13

Since $f$ is absolutely continuous we have $$f(x)-f(y)=\int_y^x f'(t)dt\geq 0, \qquad \text{ for }y<x.$$ In fact this shows that if $f'>0$ a.e. you get $f$ strictly increasing.