The last post ended with the following problem:
Problem: Find all continuous solutions to the following functional equation:
In the previous post I explained why all continuously differentiable solutions of the functional equation (FE) are linear, that is, of the form , but now we remove the assumption that the solution be continuously differentiable and ask whether the same conclusion holds. I found this problem to be extremely interesting, and at this point I will only give away that I eventually solved it, but after five (!) years.
In principle, it is plausible that, when one enlarges the space of functions in which one is searching for a solution from to the much larger , then new solutions will appear. On the other hand, the dynamical system affiliated with this problem (the dynamical space generated by the maps and on the space ) is minimal, and therefore one expects the functional equation to be rigid enough to allow only for the trivial solutions (at least under some mild regularity assumptions). In short, a good case can be made in favor of either a conjecture that all the continuous solutions are linear or a conjecture that there might be new, nonlinear solutions.