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.
Real Analysis – Continuous and Bounded Variation Does Not Imply Absolutely Continuous
absolute-continuitybounded-variationexamples-counterexamplesreal-analysis
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Best Answer
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.
See http://mathworld.wolfram.com/DevilsStaircase.html