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Smooth means has derivatives of all order, and analytic means can be given as a convergence of power series.

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    $\begingroup$ I suggest keeping this question open, because the answers are different from those for the previous question. $\endgroup$ Commented May 17, 2013 at 8:49

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A usual example is $$ f(x)=\sum_{k\in A} e^{-\sqrt{k}}\cos kx $$ where $A=\{2^n:n\in\mathbb{N}\}$.

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