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Among mathematicians there is lot of folklore knowledge for which it is not obvious how to find original sources. This knowledge circulates orally.

An example: Among math competition folks, a common conversation is the search for a function over the reals that is infinitely differentiable, with it and all derivatives vanishing only at 0. I think $$f(x) := e^{-\frac{1}{x^2}}\mathrm{\;\;\;\; for\;\;} x \neq 0$$ is an answer to this one, and it is not hard to prove.

Is there any collection of such mathematical folklore, with proofs?

See also my follow-up question here.

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Can you perhaps be a bit more precise with what kind of folklore you mean? The example you give is a very standard example that will appear in almost any first year analysis textbook. It is certainly not folklore knowledge but rather belongs to the very fundamentals of undergrad analysis. – Ittay Weiss Feb 2 '13 at 3:50
If it is written down in a repository, it's not really folklore, is it? – Chris Godsil Feb 2 '13 at 3:54
@ChrisGodsil The idea is for someone to write folklore down for common reference. – Lover of Structure Feb 2 '13 at 4:04
@IttayWeiss Thanks for your question. I must admit it has been a while since I have practiced mathematics, but the existence of mathematical folklore is well-known; Wikipedia also says something about it. It is good that you are asking this question, but I may in fact not be the best person to answer it :-) I do think that the question is a useful one. And I certainly invite anyone to improve it. What is "math training camp folklore" may very well be considered trivial canon by a real mathematician, but that's a difference in level only. – Lover of Structure Feb 2 '13 at 4:08
@IttayWeiss Another example was the "continuous everywhere but nowhere differentiable" function. Nowadays it's very easy to find examples and proofs for this on the web, but it'd be cool to have a repository for people to browse. Basically: interesting theorems or counterexamples many people have heard of ("interesting" because that's what people talk about) but that are likely not in the standard array of textbooks. – Lover of Structure Feb 2 '13 at 4:47

Examples of the type given in the comments were collected many years ago in a wonderful book by Gelbaum and Olmsted called Counterexamples in Analysis. There is also a book called Counterexamples in Topology, and there may be some other books with similar titles.

There are also many "counterexample" threads here and on MathOverflow.

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Some folklore problem solving techniques are collected at the Tricki. It is not yet very complete, but an example of a good model article is here.

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One place that contains a lot of mathematics is the nLab. It is largely centered on higher category theory/homotopy theory but also contains a lot of general stuff. It is certainly not aiming to only contain folklore knowledge but it does contain a lot of it.

Wikipedia will also certainly contain folklore knowledge embedded somewhere in the millions of pages of information but perhaps its accuracy is more questionable than what the nLab offers.

Various maths dedicated blogs will also contain folklore and anecdotes.

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