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(I already have self-taught calculus) Are there functions [tex]f:\mathbb{R} \rightarrow \mathbb{R}[/tex] that are EVERYWHERE smooth (infinitely many times differentiable [tex]\forall x \in \mathbb{R}[/tex]) but NOWHERE analytic (Taylor series does not equal f(x) for any real x. don't gimme a bump function.)? example of such a function? and how does one construct such a function?

i am thinking of something like starting with some non-smooth function f and then replace the non-smooth points with smooth but non-analytic points, and call the resulting function g, take the limit as the number of non-analytic points fills the whole real line? i don't know what i am doing... my mathematical intuition (for a lack of a better term) tells me that i am not doing it right.

i am thinking of something like starting with some non-smooth function f and then replace the non-smooth points with smooth but non-analytic points, and call the resulting function g, take the limit as the number of non-analytic points fills the whole real line? i don't know what i am doing... my mathematical intuition (for a lack of a better term) tells me that i am not doing it right.

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