" What about equation : f(m(x ,y) , x ) = f(x ,y) ? "

[""]

 Voilà ,

 f(m(x ,y) ,x ) = f(x ,y)      (1)
 m  and  f:R^2 -> R   continous for x and y .

In a first step , I  examine pairs  (f,m)  satisfying (1)
such as 1° f(x, y) = constant
              2° m(x,y) =y and  f(x, y) = f(y, x)
              3° f(x ,y) = g(x +(1 -d)*y)  ;m(x ,y) =d*x +(1 -d)*y ...

Here is an other exemple:
             f(x ,y) = g( 1/x +y^2)   ; m(x ,y) = x/(1 - x^3 +x*y^2) .

What must be the link between m(x ,y) and f(x ,y)   and
can we build more general solutions ?

Friendly , Alain .