Suppose that $f$ is continuous on $[a,b]$, $f'(x)$ exists for every $x \in (a,b),$ and $f'(x)$ integrable. Prove that $f$ is absolutely continuous.
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I believe you can adapt the proof given here, for almost the same question. – Dustan Levenstein Apr 08 '14 at 19:43
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@DustanLevenstein: How is he writing $f(b) - f(a) = \int_{a}^{b} f'(x) dx$. We have to prove it. – aaaaaa Apr 09 '14 at 01:48
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isn't that just the fundamental theorem of calculus? – Dustan Levenstein Apr 09 '14 at 02:24
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@Dustan: For that we want $f$ to be absolutely continuous, which we have to prove. – aaaaaa Apr 09 '14 at 03:37
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the version of the fundamental theorem of calculus shown here is almost what you need; here it becomes an indefinite integral because you're not assuming $f'$ exists at $a$ and $b$, so you take limits approaching the endpoints, and continuity of $f$ gets you the result of the limit. Unless you're trying to prove this version of the fundamental theorem of calculus??? – Dustan Levenstein Apr 09 '14 at 04:39
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@DustanLevenstein: correct. But, for that we need Riemann integrability of $f'$ whereas we have only Lebesgue integrability of $f'$. – aaaaaa Apr 09 '14 at 11:38