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Thread: Continuous function

  1. #1
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    Continuous function

    prove using the $\displaystyle \epsilon , \delta$ definition of continuity that $\displaystyle f(x) = x^3 - 2x^2 + 1$ is continuous at $\displaystyle x=3$

    this is the start of my solution, however i need help complting it

    $\displaystyle f(x) = x^3 -2x^2 +1$
    $\displaystyle c=3$
    $\displaystyle f(c)= 27-18+1 = 10$

    for each positive number $\displaystyle \epsilon$
    $\displaystyle |f(x) -f(c)| < \epsilon$ for all $\displaystyle x $ satisfying $\displaystyle |x-c|<\delta$

    and then i get

    $\displaystyle |x-3||x^2 + x +3|<\epsilon$ for all $\displaystyle |x-3|<\delta$


    so my question is what is the value of$\displaystyle \delta$, can i just use any random number or does it have to be specific?
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  2. #2
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    Quote Originally Posted by iLikeMaths View Post
    prove using the $\displaystyle \epsilon , \delta$ definition of continuity that $\displaystyle f(x) = x^3 - 2x^2 + 1$ is continuous at $\displaystyle x=3$

    this is the start of my solution, however i need help complting it

    $\displaystyle f(x) = x^3 -2x^2 +1$
    $\displaystyle c=3$
    $\displaystyle f(c)= 27-18+1 = 10$

    for each positive number $\displaystyle \epsilon$
    $\displaystyle |f(x) -f(c)| < \epsilon$ for all $\displaystyle x $ satisfying $\displaystyle |x-c|<\delta$

    and then i get

    $\displaystyle |x-3||x^2 + x +3|<\epsilon$ for all $\displaystyle |x-3|<\delta$


    so my question is what is the value of$\displaystyle \delta$, can i just use any random number or does it have to be specific?
    Assumming your algebra is correct (I did not check it)
    $\displaystyle \begin{gathered}
    \left| {x - 3} \right| < 1\; \Rightarrow \;2 < x < 4 \hfill \\
    \left| {x^2 + x + 3} \right| \leqslant \left| {x^2 } \right| + \left| x \right| + \left| 3 \right| < 16 + 4 + 3 = 23 \hfill \\
    \delta = \min \left\{ {1,\frac{\varepsilon }
    {{23}}} \right\} \hfill \\
    \end{gathered} $
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