Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

  1. Nov 24, 2011 #1
    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
     
  2. jcsd
  3. Nov 24, 2011 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    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].
     
  4. Nov 24, 2011 #3

    Bacle2

    User Avatar
    Science Advisor

    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.
     
    Last edited: Nov 24, 2011
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook