That's a surprise, since real analysis is part of the high school curriculum in many countries. On the other hand, the restriction on the syllabus seems to reflect on the facts that many of the recent IMO problems are rather puzzle-like, which is quite depressing.

@Joel: Does it also mean that there is an f:R to R such that f is continuous outside a set of measure 0 and such that f(f(x)) = g(x)?

I'm a novice and don't understand most of what you write but I feel like there is a ton of beautiful math in your answer.

Problem 6 of the Vietnam Team Selection Test (for the IMO) of 1985 states: Suppose a function $f: \mathbb R\to \mathbb R$ satisfies $f(f(x)) = - x$ for all $x\in \mathbb R$. Prove that $f$ has infinitely many points of discontinuity. Check the following link for the whole test if you're curious: mathlinks.ro/…

@Pete: thank you. I couldn't access the link since I don't have the barcode password. But thanks anyway.

@Kevin: thanks for your answer. Obviously I don't know anything, but maybe it will be better if I adjoin all the n-torsion points to Q and find the ring of integers in that field instead?

Thanks for your enlightening input. If you don't mind, could you paste your response to my question? or make it an entirely new question? so other people can respond to "the correct question"?

An annoying feature of this site is that a lot of the book portions do not include the preface or introduction which is usually the first thing I want to read about a book.

I'm not in academia any more and so can't access Mathscinet (and JStor). It's very frustrating.

This article is very nice, for the trick and also the exposition. Thanks!. A curiosity, who is its author? Sorry, but the author isn't listed at the site. Maybe you? :-)

A related and naive question: instead of answering when a prime p splits, has people use a similar technique to say anything about when the Frobenius $Frob_p$ has certain orders (e.g. generating the full Galois group)?

@Clark I'm not sure whether Eisenbud's book is boring, however why is it inappropriate to say so if he/she feels it's boring? It has nothing to do with the author being a nice person or not. If a nice mathematician writes a wrong paper, would it be inappropriate to say his work is wrong? (which seems to be more damning than "boring")