Give an example of a function $$f:U\longrightarrow \mathbb{R}$$ uniformly differentiable with unbounded derivative $f'$, $U\subset\mathbb{R}$ open set.
Any hints would be appreciated.
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1This seems very easy if you allow $U$ to be unbounded... – Anthony Carapetis Sep 13 '13 at 23:09
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Yes especially if you work in one dimension – Evan Sep 13 '13 at 23:24
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On the other hand if you require $U$ bounded then $f$ uniformly differentiable $\implies$ $f'$ uniformly continuous $\implies$ $f'$ extends to continuous function on the (compact!) closure of $U$ $\implies$ $f'$ is bounded. – Anthony Carapetis Sep 13 '13 at 23:35
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What do you mean by uniformly differentiable? – Yiorgos S. Smyrlis Dec 22 '13 at 09:48
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http://math.stackexchange.com/questions/240296/what-is-the-precise-definition-of-uniformly-differentiable – Pete L. Clark Dec 22 '13 at 10:40
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Take $f(x) = x^{3/2}$ on $U = [0,\infty).$ Then $f'(x) = (3/2)x^{1/2},$ which is uniformly continuous on $U.$ By the MVT, $f'$ is uniformly differentiable on $[0,\infty).$ Since $f'$ is unbounded, we have an example.
zhw.
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Let's work in one dimension: What is the simplest function which is uniformly differentiable on $\mathbb{R}$? In the definition what if you even take epsilon to be zero?
Evan
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