If f is infinitely differentiable and analytic on a dense set is f analytic?

  • Thread starter deluks917
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  • #1
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Main Question or Discussion Point

Let f: R->R. If f is infinitely differentiable and analytic on a dense set is f analytic? Is this true if we restric f to [0,1]?

note: by analytic I mean the radius of convergence of the taylor expansion is non-zero about every point.

Maybe this is simple but I was thinking about it and can't figure it out
 

Answers and Replies

  • #2
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Depending on what you want, the bump function

[tex]\mathbb{R}\rightarrow \mathbb{R}:x\rightarrow \left\{\begin{array}{cc} e^{-1/x^2} & \text{if}~x\geq 0\\ 0 & \text{if}~x=0\end{array}\right.[/tex]

is a counterexample. It is analytic on [itex]\mathbb{R}\setminus \{0\}[/itex] but it is not analytic on entire [itex]\mathbb{R}[/itex].
 
  • #3
Bacle2
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Right, any smooth function of compact support , say in [a,b], is not analytic at either of the endpoints a,b , because the series must approach different values from the left and right.
 
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