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Possible Duplicate:
Proof of a simple property of real, constant functions.

Suppose $|f(x)-f(y)|\leq (x-y)^2$ for all $x,y\in\mathbb{R}$. Show f is differentiable.

This follows intuitively, the derivative $2(x-y)$ is defined on $\mathbb{R}$. How do I show this formally?

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    "the derivative $2(x-y)$ is defined on $\mathbb{R}$" does not make any sense.2012-04-12
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    Hint: Let $x=a+h$ and $y=a$. Then $|f(a+h)-f(a)| \le h^2$.2012-04-12
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    This problem is easier than this one: http://www.artofproblemsolving.com/Forum/resources.php?c=142&cid=26&year=2007&sid=3d871034c4cf7613087eda8d1270dfa8 .2012-04-12
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    I restored the content of the question because it has two answers already, including one that is quite detailed.2012-04-12

4 Answers 4