Decreasing functions Let $u:[a,b]\to \mathbb{R}$ be a continuous and a. e. differentiable function (with respect to the Lebesgue measure).
Is it true that $u' < 0$ a. e. in $[a,b]$ implies $u$ strictly decreasing everywhere in $[a,b]$?

(New question added on 12/21/2012)
I know the answer is negative (thanks to Jonas and Cameron).
But, what appens if $u$ is absolutely continuous in $[a,b]$?
In other words, is it true that $u^\prime \leq 0$ implies $u$ decreasing in $[a,b]$ when $u$ lies in the Sobolev space $W^{1,1}(a,b)$?

Hint: Cantor-Lebesgue function minus $x$.
