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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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    If $f$ is $\alpha$-Hoelder and $\alpha>1$ then $f$ is constant. Indeed, it has to be differentiable with vanishing derivative.2011-09-27
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    Hopefully this is not an attempt to salvage your PhD thesis ( http://mathoverflow.net/questions/53127).2011-09-27
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    Consider the reals with metric $d(x,y)=\vert x-y\vert^{1/\alpha}$. There's plenty of Hölder continuous functions with exponent $\alpha$.2011-09-27
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    You may want to ask Ian Morris (http://mathoverflow.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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