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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
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Taking a theorem as a definition and proving the original definition as a theorem
An example which is not quite what is being asked, and which is certainly much less sophisticated than the other answers, but which comes up in my calculus teaching around this week each year: continu …