Alsina, Claudi; Sablik, Maciej; Sikorska, Justyna On a functional equation based upon a result of Gaspard Monge. (English) Zbl 1118.39009 J. Geom. 85, No. 1-2, 1-6 (2006). Let \(I\) be an interval contained in \(\mathbb R^{+}\). A continuous function \(f: I \to \mathbb R^{+}\) satisfies the equation \[ \left| \frac{1}{2}(y - x)f\left(\frac{x + y}{2}\right) - \frac{1}{2}\left(f(y) - f(x)\right )\frac{x +y }{2}\right| = \int_{x}^{y}f(t)\,dt + \frac{1}{2}xf(x) - \frac{1}{2}yf(y) \tag{1} \] if and only if \(f\) is of the form \(f(x) = a x + b\), where \(a \in R\) and \(b \geq 0\). A similar result is proved for the equation \[ \left| \frac{1}{2}(y - x)f\left(\frac{x + y}{2}\right) - \frac{1}{2}\left(f(y) - f(x)\right )\frac{x + y}{2}\right| = \frac{1}{2}yf(y) - \frac{1}{2}xf(x) - \int_{x}^{y}f(t)\,dt. \] Equation (1) is motivated by a classical result of Gaspard Monge. Reviewer: Andrzej Smajdor (Kraków) Cited in 1 Document MSC: 39B22 Functional equations for real functions Keywords:Monge’s theorem; affine function PDFBibTeX XMLCite \textit{C. Alsina} et al., J. Geom. 85, No. 1--2, 1--6 (2006; Zbl 1118.39009) Full Text: DOI