On Functions Taking the Same Value on Many Pairs of Points

Abstract

Let $0<p<1,\, p\neq 1/2 ,$ and let $I\subset \mathbb{R}$ be an interval. We say that the function $f:I\to \mathbb{R}$ has property ${(\rm M_p )}$ if, whenever $x,y \in I$ and $f(x)\neq f(y),$ then $f\big( px+(1-p)y\big)=f\big( (1-p)x+py\big)$. We prove that \begin{enumerate} \item[(i)] if $f$ has property ${(\rm M_p )}$ and has a point of continuity in $I,$ then $f$ is constant apart from a countable set; \item[(ii)] if $f$ is measurable and has property $({\rm M_p })$, then $f$ is constant a.e. \end{enumerate} As a corollary we obtain that if $f$ is a derivative and has property ${\rm ( M_p )}$, then $f$ is constant. Then we apply this result to solve a functional equation that appears in a variant of the Matkowski-Sut\^{o} problem.