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All of the non-analytic smooth functions I've come across so far have either been piecewise-defined or involved infinite sums. Are there any examples of non-analytic smooth functions that consist of a single expression with a finite number of operations?

If the answer is no, what theorem or lemma can be used to prove this?

Matt D
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    What's your definition of "piecewise"? Because, the way the word is generally used, any function can be "piecewise" if you wish to treat it that way. – CardioidAss22 Apr 03 '20 at 01:54
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    By piecewise I mean functions that can only be defined using distinct formulas for different sub-domains. I'm looking for non-piecewise functions: functions that can be described using a single formula for the entire domain (even if they can also be described in a piecewise manner). – Matt D Apr 03 '20 at 02:12
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    I think what I'm looking for is a function that consists of a single expression, actually. Is that clearer? – Matt D Apr 03 '20 at 02:21
  • Thinking about this further, maybe I'm just looking for closed-form expressions. Are those guaranteed to be analytic? – Matt D Apr 03 '20 at 02:42
  • $f(x)=\exp(-1/x^2), f(0)=0$ is not analytic at $0$, but I don't know if you would consider it a "closed-form" expression. – Célio Augusto Apr 03 '20 at 03:43
  • I have an active question about an example that I believe fulfills what you ar asking for here... but take caution since I am exactly asking if it is keeping their smoothness on the whole real line. Hope it helps. – Joako Feb 25 '22 at 16:14
  • By the way... a well known examples of non-analytic smooth functions are named "bump-functions", in case it helps with your search. Good luck. – Joako Feb 25 '22 at 16:16

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All elementary functions are analytic on any open set of their domain. Compositions of analytic functions are analytic. Sums, products, and quotients, of analytic functions, are also analytic (as long as the function we divide by is non-zero).

Therefore it is not possible to find a function which is smooth at a point yet non-analytic there, unless you step outside of finite compositions/sums/products/quotients of elementary functions. That is, to get a smooth non-analytic function out of elementary functions, the function which you define has to be defined 'piecewise'.

Of course there are also non-elementary functions.

Dan
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