Need for examples in measure theory For me, measure theory is a part of mathematics dealing with Lebesgue integrals and so on.

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*So, professor told us that there are functions differentiable on $(a,b)$ but whose derivative is not Lebesgue integrable over any $[a_1,b_1] \subset (a,b)$. If someone could provide example it would be great.


*If we have a function $f$ that is absolutely continuous, then it is a standard result that it is almost everywhere differentiable, and that its derivative is integrable. It could be showed by example that if function is only differentiable almost everywhere, its derivative does not need to be integrable at all(I seeked reference for this in 1.) ,so reference I seek now can we somehow have greater class of functions then the absolutely continous ones, such that they are differentiable almost everywhere and the derivative is integrable?


*If we have a differentiable function $f$ on $(a,b)$ , then it has to be some segment $[a_1,b_1]\subset (a,b)$ such that $f'$ on $[a_1,b_1]$ is Riemann integrable. Does this have to hold? If it does not, please give me a counter-example. For those wandering around, thinking those questions are trivial, just to remind that in order that Lebesgue integral exists, you need ABSOLUTE INTEGRABILITY so negative answer on this one does imply the negative on the first one.
Thanks for your answers, hints and reference!
 A: for 3) there are examples of differentiable functions with bounded derivative whose derivative is not Riemann integrable in any interval. 
The reference is a paper of Y. Katznelson and K. Stromberg. "Everywhere differentiable, nowhere monotone, functions, Am. Math. Monthly, 81 (4), (1974), 349-353. See also the link nowhere monotone for more references.
Concerning 1) I only have a partial answer. A differentiable function sends sets of measure zero into sets of measure zero. If its derivative is Lebesgue integrable on an interval, then the function is absolutely continuous in that interval. 
In the paper Sebastian Lindner  Additional properties of the measure $v_f$ Tatra Mountains Mathematical Publications. Volume 28, 2004, No. 2: 199-205.Lindner in the Example 5 Lindner constructs a function which sends sets of measure zero into sets of measure zero but whose variation is infinite in any interval. It is not exactly (1) but it is very much in the same direction. Instead of using broken lines in his construction, one could try to use more regular functions to get differentiability.
