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Math Help - show that the function is continuous?

  1. #1
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    show that the function is continuous?

    How would i show that f(x)= x^3-3 x^2+17 is continuous on [-1,1]?
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by tn11631 View Post
    How would i show that f(x)= x^3-3 x^2+17 is continuous on [-1,1]?
    there is a theorem said if "f" and "g" continuous functions then f+g is continuous

    so I'd like to split the function into three functions

    g(x) = x^3 , h(x) = -3x^2 , j(x) = 17

    I will prove the first one and leave the others for you

    given \epsilon >0 and take c \in [-1,1]

    we need to find delta depending on epsilon and x

    \mid x^3 - c^3 \mid

     \mid (x-c)(x^2+cx+c^2) \mid

    \mid (x-c)(x^2+cx+c^2) \mid \leq \mid x-c \mid \mid (x^2+cx+c^2)  \mid

    \mid x-c \mid \mid (x^2+cx+c^2)  \mid < \epsilon

    \delta \mid (x^2+cx+c^2)  \mid < \epsilon

    choose \delta = \frac{\epsilon }{(x^2+cx+c^2)}

    this delta works for any epsilon
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