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komponisto comments on Rationality Quotes January 2010 - Less Wrong
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There is a similar story -- whether true or not I don't know -- told at Oxford about Cambridge and at Cambridge about Oxford. Someone wrote a thesis on anti-metric spaces, which are like metric spaces, except that the triangle inequality is the other way round. He proved all sorts of interesting facts about them, but at the viva, the external examiner pointed out that there are only two anti-metric spaces: the empty set and the one-point set.
It is recounted that the student passed, but his supervisor was criticised for not having picked up on this earlier.
Likewise there's the story about the Princeton student defending his thesis on the set of real functions that satisfy the Lipschitz condition for every positive constant C, and being asked by an examiner to compute the derivative of such a function...
My point having been, of course, that the k-quandle story is not (necessarily) of this type.
I don't think you need to do anything as sophisticated as computing the derivative to prove that the only such functions are constant functions. Consider any distinct x_1, x_2. d(x_1, x_2) is nonzero by the definition of metric spaces. If d(f(x_1), f(x_2)) were nonzero, there would be a K small enough for the condition to be violated; therefore it must be zero for all x_1, x_2.
The humor of asking the student to compute the derivative is that one imagines the student confidently starting to answer the question, until a dawning horror rises on the student's face as the implications of the answer become evident.