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seen Jan 11 '13 at 21:34

Dec
23
awarded  Tumbleweed
Dec
16
accepted Absolutely continuous functions with derivatives in $L^p$
Dec
15
awarded  Teacher
Dec
15
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!
Dec
15
revised Absolutely continuous functions with derivatives in $L^p$
added 8 characters in body
Dec
15
comment Absolutely continuous functions with derivatives in $L^p$
@Jonas. Yes thank you!
Dec
15
answered Absolutely continuous functions with derivatives in $L^p$
Dec
15
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.
Dec
15
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...
Dec
14
awarded  Commentator
Dec
14
comment Absolutely continuous functions with derivatives in $L^p$
@Jonas. Is it important in this case?
Dec
14
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.
Dec
13
accepted “Commutativity” of integrals
Dec
13
asked “Commutativity” of integrals
Dec
12
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.
Dec
12
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 $y$?
Dec
12
revised Absolutely continuous functions with derivatives in $L^p$
added 15 characters in body
Dec
12
asked Absolutely continuous functions with derivatives in $L^p$
Dec
11
accepted How to know a function is in $L^p$.
Dec
11
comment How to know a function is in $L^p$.
Oh sorry about my confusion! Thank you!