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.
A monthly thread for posting rationality-related quotes you've seen recently (or had stored in your quotesfile for ages).