Linked pairs of additive functions (Q1689411)

The focus of this paper is the pairs of additive functions \((f,g)\) linked by the functional equation \[ xf(x)=yg(x),\qquad (x,y)\in\mathcal{C}, \] where \(\mathcal{C}\) is some specified curve. It is found that for many types of curves this relationship forces \(f\) and \(g\) to be equal and linear. However, there are some exceptions.