In this paper we investigate the functional equation \(\begin{aligned} \varphi \left( \frac{x+y}{2} \right) \left( \psi _1(x) - \psi _2(y) \right) = 0 \quad \left( \hbox { for all } x \in I_1 \hbox { and } y \in I_2 \right) \end{aligned}\) where \( I_1 , I_2 \) are open intervals of \( \mathbb {R}\) , \( J = \frac{1}{2} \left( I_1 + I_2 \right) \) moreover \( \psi _1: I_1 \rightarrow \mathbb {R}\) , \( \psi _2: I_2 \rightarrow \mathbb {R}\) and \( \varphi : J \rightarrow \mathbb {R}\) are unknown functions. We describe the structure of the possible solutions assuming that \( \varphi \) is measurable. In the case when \( \varphi \) 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.
