On functions taking the same value on many pairs of points (Q2520026)

Let \(I\subset\mathbb{R}\) be an interval. A function \(f: I\to\mathbb{R}\) is said to have the property \((M_p)\), where \(0< p< 1\), \(p\neq{1\over 2}\), if whenever \(x,y\in I\) and \(f(x)\neq f(y)\) then \(f(px+ (1- p)y)= f((1- p)x+ py)\). The authors show that if \(f\) is a measurable function with property \((M_p)\), then \(f\) is constant a.e. Also, if ``measurable'' is replaced by ``has a point of continuity in \(I\)'', then \(f\) is constant apart from a countable set. The authors then apply this result to a certain functional equation.