Suppose f is a continuous function such that f(1000)=999 and f(x) · f(f(x)) = 1 for all real x. What is f(891)?
(In reply to A Funny Fallacious Function?
I think you are missing something, Harry.
The sample function I gave is certainly continuous.
As for the other requirement, consider x = 999.5, which is on that line you are questioning. In my sample function, f(x) = (999 + 1/999)/2, which is clearly between 999 and 1/999. f(f(x)) = 2/(999+1/999). So f(x)*f(f(x)) = 1. The same is true for every point between 999 and 1000, because they all have f(x) between 1/999 and 999.
I could have drawn any sort of continuous curve connecting 999 and 1000, as long as f(x) was between f(999) and f(1000)
or consider x = 1,000,000
f(x) = 999
f(f(x)) = 1/999
So f(x)*f(f(x)) = 1
In fact, f(x)*f(f(x)) = 1 for all real x.