2025/08/23 by Tóth, Péter
#26A15 #39B22 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2508.17118
In this paper we investigate the functional equation φ( (x+y)/(2) ) ( ψ1(x) - ψ2(y) ) = 0 \hspace20mm ( for all x ∈ I1 and y ∈ I2 ) where I1 , I2 are open intervals of ℝ , J = (1)/(2) ( I1 + I2 ) moreover ψ1 : I1 → ℝ , ψ2 : I2 → ℝ and φ: J → ℝ are unknown functions. We describe the structure of the possible solutions assuming that φ is measurable. In the case when φ is a derivative, we give a complete characterization of the solutions. Furthermore, we present an example of a solution consisting of irregular Darboux functions. This provides the answer to an open problem proposed during the 59th International Symposium on Functional Equations.