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 Dec23 awarded Tumbleweed Dec16 accepted Absolutely continuous functions with derivatives in $L^p$ Dec15 awarded Teacher Dec15 comment Absolutely continuous functions with derivatives in $L^p$ @Jonas. Yes you are totally right. It is integrable on bounded intervals, and this is what I use for the proof. Sorry for my confusion! Dec15 revised Absolutely continuous functions with derivatives in $L^p$ added 8 characters in body Dec15 comment Absolutely continuous functions with derivatives in $L^p$ @Jonas. Yes thank you! Dec15 answered Absolutely continuous functions with derivatives in $L^p$ Dec15 comment Absolutely continuous functions with derivatives in $L^p$ ...is integrable, and we have $F(b)-F(a)= \int_a^b F'(x)dx$. I will post an answer to my question so that you will understand the discussion between Giuseppe and I. Dec15 comment Absolutely continuous functions with derivatives in $L^p$ @Jonas: The problem comes from a problems set that our professor gave us to practise for the final exam. I am not sure where she took them from. The definition of absolutely continuous function that I use is: For every $\epsilon>0$, there exists $\delta > 0$ such that for any finite collection of disjoint intervals $(a_i, b_i)_{i=1}^n$, $\sum_i |F(b_i)-F(a_i)|<\epsilon$ whenever $\sum_i (b_i-a_i) < \delta$. This is the definition from Folland's real analysis. The Fundamental theorem of calculus states that $F$ is absolutely continuous if and only if $F'$ exists almost everywhere and... Dec14 awarded Commentator Dec14 comment Absolutely continuous functions with derivatives in $L^p$ @Jonas. Is it important in this case? Dec14 comment Absolutely continuous functions with derivatives in $L^p$ @Jonas. It is not directly part of my definition, but it follows from the fundamental theorem of calculus for Lebesgue integration. Dec13 accepted “Commutativity” of integrals Dec13 asked “Commutativity” of integrals Dec12 comment Absolutely continuous functions with derivatives in $L^p$ Oh if you take $L = \int_{\mathbb R} f'(t) dt$, then I think it works applying Holder. Dec12 comment Absolutely continuous functions with derivatives in $L^p$ I thought about that. Then doesn't the constant $L$ that you get depend on $x$ and $y$? Dec12 revised Absolutely continuous functions with derivatives in $L^p$ added 15 characters in body Dec12 asked Absolutely continuous functions with derivatives in $L^p$ Dec11 accepted How to know a function is in $L^p$. Dec11 comment How to know a function is in $L^p$. Oh sorry about my confusion! Thank you!