Identify a real valued function f(x) which has the property for all real x that f(f(x)) = -x.

David Karr presents one solution:

> Define [x] = ceil(x) = the least integer greater than or equal to x.
>
> f(0) = 0.
> If x > 0 and [x] is odd, then f(x) = 1 + x.
> If x > 0 and [x] is even, then f(x) = 1 - x.
> If x < 0, then f(x) = -f(-x).

He then makes some useful observations and asks:

> Are there any other interesting facts we can prove about f ... ?

Well, for those who find such things interesting, here is a complete
characterization of functions with the desired property:

Call a set G of ordered pairs of real numbers a GENERATOR if it
satisfies:

    (A) If (u,v) in G then 0 < u < ABS(v)

    (B) If r > 0 there is exactly one (u,v) in G such that
        r = u or r = ABS(v)

In other words, pair off the positive reals.  Then go through the pairs
and at whichever of them you please replace the larger member of the
pair with its negative.  e.g.  (1, -9), (5, 17.6), (3, 26),
( pi, -SQRT(97) ), (3/7, 29), etc.

A real function f is GENERATED by a generator G if it satisfies:

    (C) f(0) = 0

    (D) For any (u,v) in G: f(u) = v, f(-u) = -v, f(v) = -u, f(-v) = u

It is easy to see that any generated function has the desired property
that f(f(x)) = -x for all x.

Nor is it very difficult to show that every such function is generated
by a generator.  We leave that part as an exercise.

-- Barry Paul <barry.paul@hofbbs.com>, Great Neck, NY