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Does there exist a pathological function which is differentiable any finite number of times as one wishes, but is not differentiable an infinite amount of times?

Is it reasonable for such function to exist?

Joshhh
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    What do you mean by "differentiable an infinite number of times", if not "differentiable $N$ times, for all $N$"? – Patrick Stevens Feb 07 '16 at 11:34
  • The motivation for this question comes from similar weird things from set theory. Like, without the axiom of choice, any finite family of sets has a choice function, but when you "cross" the border to infinite, this is not generally the case. – Joshhh Feb 07 '16 at 11:36
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    So what does it mean to differentiate an infinite number of times? – Patrick Stevens Feb 07 '16 at 11:36
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    People tend to give meaning to words without thinking about it. There's no such thing as the concept of "infinitely differentiable", not unless one uses the term as an abbreviation of "differentiable $n$ times,for all natural numbers $n$" - and in this case the question makes little sense. – Git Gud Feb 07 '16 at 11:41
  • There are questions on this topic here and here. – TonyK Feb 07 '16 at 11:58

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When you say infinite differentiation I imagine limit $$\lim_{n\rightarrow+\infty}f^{(n)}$$

I'm not sure if such definition exists, it's just what I imagine. So for example $$f(x)=e^{-x}$$ $$f^{(n)}(x)=(-1)^ne^{-x}$$ But this sequence doesn't converge - so if you're looking for a function, that is $n$ times differentiable $\forall n$, but the limit does not converge, this is one example and I'm sure there are many more.

Tom83B
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