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Suppose $f$ is a Hölder continuous function on $\mathbb R$ with exponent $\alpha >1$. It can be proved that it has to be zero.

But, are there other spaces on which nontrivial Hölder continuous functions can be defined and are nontrivial? Are there interesting applications for such a line of thought?

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    You may want to ask Ian Morris (http://mathover$f$low.net/users/1840/ian-morris), who mentioned studying such spaces in a comment on the answer @RagibZaman linked: http://mathoverflow.net/questions/53122/mathematical-urban-legends/53127#comment132138_531272015-01-30

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