An elementary method of solving functional equations

Abstract. In the present paper we prove in an elementary way that if for a fixed \(\lambda\in (0,1)\) the functions \(f, g:\mathbb{R}\longrightarrow \mathbb{R}\) satisfy the equation $$ \frac{f(b)-f(a)}{b-a} = g\left(\lambda a + (1-\lambda) b\right) $$ for all \(b>a\) then \(f\) is a quadratic polynomial, and \(g=f'\). Moreover, if \(\lambda \neq \frac{1}{2}\), then \(f\) is a linear polynomial and \(g=f'\). This result is obtained with no regularity assumptions on \(f\) or \(g\) and generalizes a theorem from [4].