Okay, so after you play around with it for a bit, you can realize, much to your annoyance, that the solutions are something like and , which are just really stupid solutions for a functional equation.

So let's substitute to get a functional equation with nicer-looking solutions. Unfortunately this means there will be a lot of mundane term-canceling in the background that will make following this proof without any of your own manipulation extremely annoying. At least, that's how I feel about it. Meh, algebra.



Now for some simple stuff. First, quickly yields . Then, yields from which we have .

If we have then yields for all real , which obviously implies , which is just as obviously a solution, yielding

So now we may assume .

Now what we really want is to prove that , so for now let's set and see what happens.

yields . In particular with we find so .

yields . In particular . Combining with the above yields .

Hmm, this is annoying. Could there just be some weird sort of oscillating fractional-part solution in this innocuous-looking functional equation? Oh snap. Let's keep going.

For now, let's suppose that . Then yields , and yields

Also, , which, plugging back, yields for all , which, although being a solution, contradicts . Whew, no sporadic monsters. Yay, .

Now... yay! In fact, we even have now! No, let's be even more epic: by induction we have that for any positive integer , and as a trivial special case, ! This also holds for negative integers, just as obviously, by going the other way.

And now if we take to be an integer and plug back with we get .

Yes, we're close. Since yields , we can get to give us .

Now we just make this functional equation blow up with the archimedean method. Suppose, for some evil number , we have ; we can pick a freaking large-magnitude integer so that is at least 1, then take an integer just under to get is positive yet is negative, which contradicts .

And therefore the only other solution is , yielding . Oh, and it's easily checked to satisfy the original functional equation. (<-- This statement is worth 1 point out of 7. QQ)