We consider continuous solutions f: R+→ R+=(0,∞) of the functional equation f(xf(x)) =φ(f(x)) where φ is a given continuous map R+ → R+ without any restrictions. It can be treated as a difference equation but the standard methods lead only to (usually very irregular) discontinuous solutions. Concerning continuous solutions, the case when φ is an increasing homeomorphism, is completely described, in the other cases there are only partial results. This paper contains a survey of known results, new results showing how the equation can be simplified in the general case, and some open problems and conjectures. In particular, can the range of a solution contain a periodic orbit of φ of period different from 1 and 2? We provide auxillary results that could be of some use. Finally, this equation was also considered in the complex domain and some classes of entire and/or locally analytic solutions are described. Again, there are open problems and conjectures here.