How to show that every bounded variation function on $[a,b]$ is differentiable a.e?

Question

I'm struggling with this question for over a week now. I know the proposition is true, but haven't managed to prove it yet. any suggestions anyone? ($f$ is BV on $I$ if $$\sup\left\{\sum|f(b_k)-f(a_k)| :a_{k+1}\gt b_{k}\gt a_{k} ; a_k,b_k\in I\right\}\lt \infty)$$

Are you allowed to use the fact that monotonic functions are differentiable a.e.? — Etienne, Aug 28 '13 at 12:34

score 4 · Accepted Answer · answered Aug 28 '13 at 12:46

4

Every function of bounded variation is the difference of two monotonically non-decreasing functions. Monotonically non-decreasing functions are differentiable almost everywhere.

answered Aug 28 '13 at 12:46

robjohn

353,833

please prove both these claims – JustSomeGuy Aug 28 '13 at 12:47
1

@Snir: These results are quite standard. For example, you can find them both proven in Prof. Tao's book "An Introduction to Measure Theory", which can be found for free online (just ask Google!) – Elchanan Solomon Aug 28 '13 at 12:54
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@Snir, see also Michael W. Botsko, An Elementary Proof of Lebesgue's Differentiation Theorem, The American Mathematical Monthly, 110 (2003), 834-838. – Umberto P. Aug 28 '13 at 12:54
Snir didn't say his function is real-valued. If it is complex valued: first show the real and imaginary parts are BV, then use the result here. – GEdgar Aug 28 '13 at 13:39

How to show that every bounded variation function on $[a,b]$ is differentiable a.e?

1 Answers1

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