On solutions of the Fréchet functional equation (Q884359)

The paper begins with a survey on the Fréchet equation \(\Delta_h^n f=0,\) where \(f: \mathbb{R}\to\mathbb{R}\), \((\Delta_h^1 f)(x)=f(x+h)-f(x),\) \(\Delta_h^{n+1} =\Delta_h^1(\Delta_h^n),\) \( x\in\mathbb{R},\) \(h\in]0,\infty[,\) \(n\in \mathbb{N},\) its particular cases the Cauchy and Jensen equations, its generalizations where there are \(h_1,h_2,\dots\) in place of just one \(h\) or \(x_1,x_2,\dots\) in place of \(x.\) Then the authors offer new proofs of results on these equations and results on the graphs of their solutions.