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Math Help - Continuous function

  1. #1
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    Continuous function

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

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

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

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

    and then i get

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


    so my question is what is the value of \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 \epsilon , \delta definition of continuity that  f(x) = x^3 - 2x^2 + 1 is continuous at x=3

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

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

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

    and then i get

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


    so my question is what is the value of \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)
    \begin{gathered}<br />
  \left| {x - 3} \right| < 1\; \Rightarrow \;2 < x < 4 \hfill \\<br />
  \left| {x^2  + x + 3} \right| \leqslant \left| {x^2 } \right| + \left| x \right| + \left| 3 \right| < 16 + 4 + 3 = 23 \hfill \\<br />
  \delta  = \min \left\{ {1,\frac{\varepsilon }<br />
{{23}}} \right\} \hfill \\ <br />
\end{gathered}
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