
Re: Existence of a function
Posted:
Aug 15, 2013 4:14 PM


> Does there exist a function f: R > R such that > f(f(x)) =/= x for all x in R and > > for every a in R there exists a sequence {x_n} such > that x_n > f(a) and f(x_n) > a as n>oo ? > > > Message was edited by: Pastamak
Probably yes, but I don't have a construction. Obviously f cannot be continuous. I will work with (0,1) rather than R. We would like a set P of points in (0,1)x(0,1) {(X,Y)} = {(X,f(X)} such that P is dense in (0,1)x(0,1), each X corresponds to exactly one f(X), and f(f(X)) =/= X. The existence of spacefilling curves tells us it won't be hard to make P dense, and I suspect a "random" function f would work, after the occasional violation f(f(x))=x is taken care of. It would be better to construct a concrete f, but I don't see it offhand.
Once we have P, for a given A in (0,1), find a sequence of points (x_n, f(x_n)) in P converging to (f(A),A).
Don Coppersmith

