Is there a good repository for mathematical folklore knowledge? 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.
 A: 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. 
A: 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.
A: 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.
