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Can someone give me a reference where I can see a proof that a power function of some exponent between 0 an 1 is Hölder continuous with the same exponent in some compact set? I have seen a trick to prove it in the case of exponent $\frac{1}{2}$, but I would like to see how to prove it in the general case.

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Could you make a more precise statement? –  Pietro Majer Feb 19 '12 at 21:17
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If you just mean why $f(t):=|t|^\alpha$ with $\alpha\in (0,1)$ is $\alpha$-Hölder, the reason is that $f$ is concave, hence sub-additive, hence it is a modulus of continuity of itself (check en.wikipedia.org/wiki/Modulus_of_continuity) –  Pietro Majer Feb 19 '12 at 21:18
    
It seems the e-mail notification is not working, so I have returned here only now. Thanks for the clue. Actually, after reading it I was thinking again about an elementary proof, and indeed the key is the subadditivity of the function. Using it, the required result is no more difficult to prove than $||x|-|y|| \leq |x-y|$ from the triangle inequality. –  António Caetano Feb 24 '12 at 0:52

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