Geodesic
New examples of Pompeiu functions
Eurasian mathematical journal, Tome 4 (2013) no. 3, pp. 63-69.

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For given sequence of real numbers $\{x_k\}^\infty_1\subset I:=[0,1]$ the explicitly defined function $\varphi\colon I\to I$ is constructed such that $\varphi(x_k)=0$, $k\in\mathbb N$, $\varphi(x)>0$ a.e. and all $x\in I$ are Lebesgue points of $\varphi(\cdot)$. So its primitive $f(\cdot)$ is an everywhere differentiable strictly increasing function with $f'(x_k)=0$, $k\in\mathbb N$. 
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G. A. Kalyabin. New examples of Pompeiu functions. Eurasian mathematical journal, Tome 4 (2013) no. 3, pp. 63-69. http://geodesic.mathdoc.fr/item/EMJ_2013_4_3_a5/

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