torquestomp
New member
Let H be the set of all functions h for which there exists a continuously differentiable real function f such that h(x) = f(f(x)). Such functions include h(x) = x, h(x)=e^(e^x), etc. (H is clearly infinite in size).
What can we derive in general about h? Are there any non-trivial operations under which H is closed? What are we describing if we extend H to transformations in R^n?
Has H been studied at all? Or is it just a cramped, relatively pointless set of functions?
I am not looking for any specific answers here, just some discussion with luck some enlightening deductions. However, for those who fear these kinds of questions, here are some more specific questions (more food for thought than anything. These have relatively trivial proofs):
a) Prove that if h is bounded if f is bounded, but f unbounded does not force h unbounded
b) Prove that f is not unique, even when h(x) constant
c) Prove that h(x) = -x is not a member of H
* a) Prove that h is bounded if f is bounded, but that h may still be bounded even if f is unbounded
What can we derive in general about h? Are there any non-trivial operations under which H is closed? What are we describing if we extend H to transformations in R^n?
Has H been studied at all? Or is it just a cramped, relatively pointless set of functions?
I am not looking for any specific answers here, just some discussion with luck some enlightening deductions. However, for those who fear these kinds of questions, here are some more specific questions (more food for thought than anything. These have relatively trivial proofs):
a) Prove that if h is bounded if f is bounded, but f unbounded does not force h unbounded
b) Prove that f is not unique, even when h(x) constant
c) Prove that h(x) = -x is not a member of H
* a) Prove that h is bounded if f is bounded, but that h may still be bounded even if f is unbounded