On the stability of the equation stemming from Lagrange MVT (Q617029)

The functional equation \[ F(y) - F(x) = (y-x) \, f\left ( {{x+y} \over 2} \right ) \] for all \(x, y \in \mathbb{R}\) arises in connection with the Lagrange mean value theorem in analysis. Solution of this and many other related functional equations are contained in the book ``Mean Value Theorems and Functional Equations'' by the reviewer and \textit{T. Riedel} [World Scientific, Singapore (1998; Zbl 0980.39015)]. In this paper, the authors prove that this functional equation is super stable. In particular, they prove the following results: If the functions \(F, f : \mathbb{R} \to \mathbb{R}\) satisfy the functional inequality \[ \left | F(y) - F(x) - (y-x) f \left ( {{x+y} \over 2} \right ) \right | \leq \epsilon \] where \(\epsilon\) is some fixed positive number, then there exist constants \(c, b \in \mathbb{R}\) such that \(f(x) = c x + b \) for all \(x \in \mathbb{R}\).