I am learning analysis and all the uniformly continuous functions I have seen are over a closed interval. So, can a uniformly continuous function be defined on an open interval?
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Sure it can, a simple example is the function $f(x)=x$ on the interval $(0,1)$. You should try to rigorously prove why this is indeed uniformly continuous – Moss May 21 '13 at 06:19
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1Hmmm... $f(x)=0$ for every $x$. – Did May 21 '13 at 06:23
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possible duplicate of Absolute continuity on an open interval of the real line? – Lord_Farin May 21 '13 at 06:55
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Uniformly continuity definition doesn't depends of domain, just restrict it. – Gaston Burrull May 21 '13 at 07:24
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This result may help you:
Let ${F}:(a, b)\rightarrow \mathbb{R}$ that is continuous on the bounded open interval $(a, b)$ then the two limits given by
$F(a +) = \lim_{x\to a^{+}} F(x)$, $F(b -) = \lim_{x\to b^{-}} F(x)$ exists iff $F$ is uniformly continuous on $(a, b)$.
This result has been given in the book "The calculus integral by Brian S. Thomson".
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Sure, take a function which is continuous on a finite closed interval, and remove the endpoints.
And there are many functions defined on the whole real line that are uniformly continuous.
André Nicolas
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Do you mean remove the endpoints of a continuous function or a uniformly continuous function? – Jazz May 21 '13 at 06:35
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A function continuous on a closed bounded interval is uniformly continuous in that interval. – André Nicolas May 21 '13 at 06:37
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Just an oversight. Sorry to be so competitive, but I'll bet I have had many more of them than you. – André Nicolas May 21 '13 at 07:00