I've been stumped by this problem:
Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that $$f\circ g=h$$ $$g\circ h=f$$ $$h\circ f=g$$ or prove that no three such functions exist.
I highly suspect, by now, that no non-trivial triplet of functions satisfying the stated property exists, but I don't know how to prove it.
How do I prove this, or how do I find these functions if they do exist?
BONUS POINTS: The functions should also be continuous.