I know that a continuous function which is a BV may not be absolutely continuous. Is there an example of such a function? I was looking for a BV whose derivative is not Lebesgue integrable but I couldn't find one.
Continuous and bounded variation does not imply absolutely continuous
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real-analysis
examples-counterexamples
bounded-variation
absolute-continuity
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0My apologies; I only saw Byron's when I posted that comment. – 2010-09-15
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0See also" http://math.stackexchange.com/questions/499101/example-of-a-function-that-has-the-luzin-n-property-and-is-not-absolutely-cont – 2016-04-21
2 Answers
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The Devil's staircase function does the trick.
Its derivative is almost surely zero with respect to Lebesgue measure, so the function is not absolutely continuous.
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7This is indeed the most standard example of a function which has BV but is not AC. It might be helpful to the OP to know that such a function is (more?) commonly referred to as the Cantor function: http://en.wikipedia.org/wiki/Cantor_function. – 2010-09-14
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Byron already answered your main question, but your last sentence is another matter. You want a BV function whose derivative is not integrable, but such things don't exist. In particular, if $f$ is monotone on $[a,b]$, then $f'$ exists a.e., is Lebesgue integrable, and $\int_a^b f' \leq f(b)-f(a)$. Thus half of the fundamental theorem of calculus holds, so to speak. General BV functions are differences of monotone functions, so their derivatives are also Lebesgue integrable.
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0Thanks for pointing out my error. – 2010-09-14