On affine functions with respect to some means (Q623425)

Let \(I\) denote a non-degenerate interval of real numbers. For \(u,v\in I\) and a function \(M:I^{2}\rightarrow I,\) define the functions \(M_{u},M^{v}:I \rightarrow I\) by the formulas: \[ M_{u}(y):=M(u,y),y\in I;M^{v}(x):=M(x,v),x\in I. \] The main result of the paper is the following. Let \(M,N:I^{2}\rightarrow I\) be functions such that for all \(u,v\in I\) the mapping \(M_{u}\) is strictly increasing, the mapping \(M^{v}\) is strictly increasing and continuous, and the mappings \(N_{u},N^{v}\) are continuous. Assume that a triple \(\left( f,g,h\right) :I\rightarrow I^{3}\) is a solution of the functional equation \[ M(f(x),g(y))=h(N(x,y)),x,y\in I. \] If there exists a subinterval \(I_{0}\subseteq I\) such that \(f\) is nonconstant on \(I_{0}\) and \(f(I_{0})\subseteq [ s,S]\subseteq I,\) then \(g\) is continuous. As a consequence, it is also proved that if \(M\) is a symmetric, strict mean such that the functions \(M_{u},M^{v}\) are continuous and strictly increasing for every \(u,v\in I\) and \(f\) is an \(M-\)affine function, that is \[ M(f(x),f(y))=f(M(x,y)),x,y\in I, \] then the local boundedness of \(f\) from one side by an element from \(I\) implies its continuity.