New Examples of Pompeiu Functions

Abstract

For given sequence of real numbers \( \{x_{k}\}_{1}^{\infty} \) \(\subset\) I:=[0,1] the explicitly defined function \( \varphi:I \rightarrow I \) 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 \( f^{\prime}(x_{k})=0 \) \( k \in \mathbb{N} \).