On a functional equation based upon a result of Gaspard Monge (Q860219)

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.