New examples of Pompeiu functions

For any sequence {gamma _{k}} _{kin Bbb{N}} of positive numbers satisfying the condition {inf_{kin Bbb{N} }( gamma _{k+1}-gamma _{k}) >0}, and any countable set { x_{k}} _{kin Bbb{N}} that is dense in {I=[ 0,1]}, the author defines the function varphi :Irightarrow I by the formula varphi(x) =inf_{kin Bbb{N}}vert x-x_{k}vert ^{1/gamma _{k}}. He proves that varphi(x) >0 a.e. on I, that varphi is continuous at x if and only if varphi (x) =0, and that every xin I is a Lebesgue point of varphi . Thus f(x) =int_{0}^{x}varphi(t),dt is a differentiable, strictly increasing function with f^{prime }(x) =varphi (x) for xin I. par The author also shows that for any countable dense set Xsubset I and any pair of countable sets Y^{+},Y^{-}subset I such that X, Y^+ and Y^- are pairwise disjoint, there is a differentiable function g such that vert g^{prime}(x) vert leq 1 for every xin I, g^{prime}(x)=0 on X, g^{prime}(x)>0 on Y^{+}, and g^{prime}(x)<0 on Y^{-}.