New examples of Pompeiu functions

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 $f'(x_k)=0$, $k\in\mathbb N$.