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When I teach second-semester calculus I usually discuss the function $f$ defined by $ f(x)=e^{-1/x^2} $ for $x \neq 0$ and $f(0)=0$. Or, almost the same example, $g$ defined by $ g(x)=e^{-1/x^2} $ for $x>0$ and $g(x)=0$ for $x \in (-\infty,0]$. Both $f$ and $g$ are too smooth at $x=0$ to be analytic on a neighboorhood of the origin. Each has trivial Taylor series at $x=0$ and yet each function is clearly nonzero in any open nbhd which contains zero. So far as I'm aware, these are the standard examples to clarify the distinction between smooth and analytic functions on $\mathbb{R}$

Question: are there other examples of functions which are smooth but non analytic? Is it possible to give a smooth function which fails to be analytic on an interval? How special are the examples I offer?

I realize you can shift my examples vertically, horizontally, rescale or even add an analytic function to make it look different. Ideally I'm looking for a genuinely different looking example then the two I offered; also, given the intended audience, I dream of a formula which is accessible to calculus II students. I hope the spirit of the question is clear.

Thanks in advance for your insights!

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    There is a nice example on the wikipedia page: http://en.wikipedia.org/wiki/Non-analytic_smooth_$f$unction#A_smooth_function_which_is_nowhere_real_analytic2012-09-02

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Very nonstandard: the real function $f(x)$ on all of $\mathbb R$ with $ f(0), \; f'(0) = 1, \; f(f(x)) = \sin x . $ See https://mathoverflow.net/questions/45608/formal-power-series-convergence/46765#46765

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    I asssume you mean my gmail email. If so, still no sign of it at the moment, even in spam...2012-09-02
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If you teach the Cauchy-Hadamard formula, the example from this previous answer of mine might be decent.

PS: I'm not entirely sure I applied it correctly in the example. Compute the radius of convergence yourself to make sure.

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    You may prefer the alternative definition as given in this link: http://www.math.osu.edu/~edgar.2/selfdiff/2012-09-03