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Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where

  • $f$ is real-analytic on the open interval $(0,1)$
  • $f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that $-C\leq f(x)\leq C$ for $x\in[0,1]$).

Is it true that there is a real-analytic continuation of $f$ to the interval $[-\epsilon, 1+\epsilon]$ for some small positive $\epsilon$? If not, what conditions can be added to make it true?

Suggestions for books (or other references) where I could have learned to answer this myself would also be appreciated.

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I suggest to reask this question on math.stackexchange.com a similar site but with a broader scope. Here I vote to close. –  quid Sep 21 '11 at 15:03
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While others have provided quite smooth counterexamples, I note that your conditions do not even force $f$ to be continuous at the end-points. –  Emil Jeřábek Sep 21 '11 at 15:33
    
@quid I wasn't aware of the difference between math.stackexchange and mathoverflow. Now that I've looked that up, I agree, this question would have been better on math.stackexchange. –  Essex Sep 21 '11 at 16:35
    
Essex, thanks for the response. No problem, this is a frequent phenomenon. –  quid Sep 21 '11 at 16:39
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closed as off topic by quid, Emil Jeřábek, Qiaochu Yuan, Andreas Blass, GH from MO Sep 21 '11 at 17:29

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2 Answers

up vote 3 down vote accepted

$f(x) = \sqrt{1-x^2}$ is real analytic on $(-1,1)$, bounded and continuous on $[-1,1]$, but of course not even one-sided differentiable at the endpoints. But then Igor's example is one-sided differentiable of all orders at the endpoint $0$, but still not real analytic in any neighborhood of $0$.

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What about $e^{-1/x^2}?$

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@quid: I will probably still be cryptic, but you do agree this answers the question as stated, no? –  Igor Rivin Sep 21 '11 at 16:06
    
@quid: I agree that this is not a research question, but the police work gets really tedious, especially when it takes thirty seconds to answer the question. –  Igor Rivin Sep 21 '11 at 16:36
    
@quid: I probably voted to close about 10 questions today. –  Igor Rivin Sep 21 '11 at 16:36
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This is perhaps more appropriate for a meta thread but: 1. It is not just a question of the time it takes to answer it, it is also a question of teaching people to not expect to receive answers to inappropriate questions. 2. If one decides to answer it anyway then I think this kind of answer is quite appropriate. If it really is after all a research level question then this is a research level answer, the OP should be able to fill in the details. –  Torsten Ekedahl Sep 21 '11 at 19:39
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@IR: ok, let us agree to disagree. @TE: it's not only that the answer is terse, it is imo in a strict sense incomplete (the other one too, btw). How does it address (on whatever level) the second question (for conditions)? Except in such a cryptic way that I'd say 'the obvious ones' is hardly less cryptic. If one wishes to be brief, that's fine; but one could have been almost as brief while being complete. I see no good reason for not doing this. –  quid Sep 21 '11 at 20:32
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