# Problem about absolute continuity of a function

$f:\mathbf{R} \to \mathbf{R}$ is an increasing function with $\lim_{x\to -\infty}f=0$ ,$\lim_{x\to \infty}f=1$, and $\int_{R}f'=1$. Prove that $f$ is absolutely continuous on every interval $[a,b]$. Any help is appreciated.

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If $f$ is merely integrable, there is no guarantee that $f'$ exists almost everywhere. And even both existence and integrability of $f'$ given, we are still unable to deduce the absolute continuity of $f$. – Sangchul Lee Aug 4 '11 at 18:08
I do no know any integrable function $f$ satisfying your two limit conditions. – GEdgar Aug 4 '11 at 18:49
I am sorry, it was a misprint that the function should be integrable, what we have is that the function is increasing. – user14272 Aug 5 '11 at 2:07

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