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Math Help - Questions about Class

  1. #1
    Junior Member
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    Questions about Class

    So I'm teaching myself from a MultiVariable Calculus text and there was an extremely brief definition of the class of a function. I understand the basic concepts of the notation but had a few questions that weren't addressed by the books one paragraph definition. They are as follows:

    (1) Does the notation imply: C^{n} \subset C^{n+1} \subset C^{\infty}? So that a function of class infinity could be represented as class n?

    (2) While a polynomial function, f(x ) is of class C^{n}, such that D^{n+1}f = 0, is it's class capped at n? or is it considered of class infinity since a derivation of zero exists and is zero?
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  2. #2
    Senior Member
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    For (1), you have written it backward, but otherwise your understanding is correct:

    C^\infty \subset C^{n+1} \subset C^n

    For (2), your understanding is also correct. The function f would be a member of class C^\infty. In fact, every polynomial function has the property of being a smooth function, which is another way of saying that it belongs to class C^\infty.
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