If $f: [0,1] \to \mathbb{R}$ is a (not necessarily continuous) function satisfying
$$ |f(x)-f(y)| \leq M|x-y|^{\alpha} $$
where $M$ and $\alpha$ are fixed real numbers and $\alpha > 1$. Classify all such functions $f$.
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think derivative – Thomas Apr 21 '14 at 07:01
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That's what I was told to do but I can't seem to figure this out – Michael Strober Apr 21 '14 at 07:01
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No derivative is needed--as was explained to answer some previous questions on the site that this one is a duplicate of. – Did Apr 21 '14 at 09:22
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for $x = a$ and $y = a + h$, then: $\left|\dfrac{f(a+h) - f(a)}{h}\right| \le M\cdot |h|^{\alpha - 1}$, and letting $h \to 0$ we have: $|f'(a)| = 0$, implying $f'(a) = 0$ so $f$ must be a constant function.
DeepSea
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1@ellya: LAcarguy just show that these functions are all differentiable and has derivative 0. – Apr 21 '14 at 08:41
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