About Period 2 Points of Continuous Invertible Functions
It is often presented without proof in high school or junior college level mathematics classes the following ``theorem''.
“Theorem”. If f is a function, then f(x)=f−1(x) if and only if f(x)=x.
Taken literally, this is of course complete nonsense, and is trivially seen to be false: take for example f(x)=1/x. For an extended period of time I was under the impression that this ``theorem'' could not be rescued regardless of how many additional restrictions we impose on f, mostly because no obvious proof has ever jumped out at me. Yet after deliberating some more about this, I have failed to find a counterexample in sufficiently nice functions---to which the ``theorem'' above is usually applied to. So I became more convinced that the theorem might indeed be true under certain reasonable conditions, and eventually it become apparent that there was an easy proof in a simple case.
Theorem. Let f:[0,1]→[0,1] be a continuous bijection satisfying f(0)=0 and f(1)=1. Then f(x)=f−1(x) if and only if f(x)=x.
Proof. The equation is the same as f(f(x))=x. Let G be the graph of f. If f(x)=k and f(k)=x, then both (x,k) and (k,x) are in G. We assume x≠k by way of contradiction, and suppose without loss of generality that x<k, then (x,k) lies above the line y=x while (k,x) lies below it. So f is decreasing on a subinterval of [x,k] but increasing on a subinterval of [0,x], contradiction with the fact that it is monotonic (which is implied by the fact that it is a continuous bijection).
Of course, the assumption f(0)=0,f(1)=1 and the restriction of the domain and range of f to [0,1] are arbitrary and this theorem applies more generally. All we really need is for the function f:(a,b)→R to be injective and continuous with limx→af(x) and limx→bf(x) converging to a finite value (which is equivalent to boundedness, since f is monotone).
Interestingly this implies that a function in the conditions of the theorem above cannot admit proper period 2 points, i.e. points x so that f(f(x))=x but f(x)≠x. For which n, then, can such a function f admit proper period n points (i.e. fn(x)=x but fk(x)≠x for any 0<k<n)? This sounds like something that would be of interest to dynamical systems problems, where fixed points and questions of stability under a certain map are often of interest; most likely this problem has been fully answered already somewhere out there, though I'll have to look further to find out for sure.
