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.