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Homework Help: Proof that a function is continuous

  1. Jul 21, 2011 #1
    Prove that the function is continuous when f(x)=0
    f(x)=x4-7x3+11x2+7x-12


    f(c)-[itex]\epsilon[/itex]<f(x)<f(c)+[itex]\epsilon[/itex]

    Limits maybe taken, however, we do not have the value for c in the limit equation.
     
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  3. Jul 21, 2011 #2

    SammyS

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    It looks like you have to solve for x, when f(x) = 0, i.e., find the zeros of f(x).

    For one of the zeros, notice that the sum of the coefficients of f(x) is zero. Therefore, f(1) = 0.

    So you know that one of the factors of f(x) is (x-1). Use long division or synthetic division to find g(x) such that: f(x) = (x-1)g(x).

    Added in Edit.
    Notice that: f(-x) = x4+7x3+11x2-7x-12. Therefore, f(-(1)) = 0 .
     
    Last edited: Jul 21, 2011
  4. Jul 21, 2011 #3
    Solving this gives (x-1)(x+1)(x-4)(x-3)
    But is this sufficient to show that it is continuous?
     
  5. Jul 21, 2011 #4

    SammyS

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    Of course not.

    The problem is to show that f(x) is continuous when f(x)=0. So the problem has become: show that f(x) is continuous for x = -1, 1, 3, 4 .
     
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