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Does there exist function $ f $ that is smooth but not analytic for all $x\in \Bbb{R}$? ($f $ is a Real-valued function)

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$$\phi(x) = e^{-1/(1-x^2)} 1_{|x|< 1}\in C^\infty_c(\Bbb{R}), \qquad \Phi(x) = \sum_{k=0}^\infty e^{-e^{e^k}}\sum_{n=-\infty}^\infty \phi(2^k x+n) \in C^\infty(\Bbb{R})$$

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