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Math Help - Smooth function

  1. #1
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    Smooth function

    Someone can help me attack this problem?

    Let e: R into R by
    e(x) = e^ (-1/x) if x > 0 and e(x) = 0 if x < or equal 0.
    1) Show that e is smooth, that is, e has derivatives of all orders at all points x . Hint use L'Hospital and induction
    2) Is e analytic?
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  2. #2
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    No, it is not analytic. Because all derivatives are zero. Thus, it would mean f converges to the zero power series on any interval.
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