A characterization of some classes of functions F of the form \(F(x,y)=g(\alpha f(x)+\beta f(y)+\gamma)\) or \(F(x,y)=\phi (h(x)+k(y))\) (Q1057433)

The authors give necessary and sufficient conditions for continuous functions on the Cartesian square of a closed subinterval [a,b] of [- \(\infty,\infty]\), nondecreasing in each variable, to be of one of the forms in the title, where \(\phi\),h,k,g,f are continuous, \(f(a)=0\), f,h,k increasing, \(\phi\) initially increasing then constant, \(g(x)=a\) for \(x\leq f(a)\), \(g(x)=f^{-1}(x)\) if f(a)\(\leq x\leq f(b)\) and \(g(x)=b\) for \(x\geq f(b)\), furthermore \(\alpha >0\), \(\beta >0\), \(\gamma\geq 0.\) \{On page 140, in the fourth paragraph a ''Remark V after theorem 2'' is quoted. This reviewer did not succeed to find this remark.\}