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Thread: Continuity Question

  1. #1
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    Continuity Question

    Let F(X) =

    5x-4, if x is less than or equal to 2

    -4x+b, if x is greater than 2

    If is a function which is continuous everywhere, then we must have b=_______?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by redraider717 View Post
    Let F(X) =

    5x-4, if x is less than or equal to 2

    -4x+b, if x is greater than 2

    If is a function which is continuous everywhere, then we must have b=_______?
    intuitively, what does continuous mean? it means there are no gaps, which means, we want -4x + b to pick up exactly where 5x - 4 left off.

    let $\displaystyle f_1(x) = 5x - 4$ and $\displaystyle f_2(x) = -4x + b$

    we want, $\displaystyle f_1(2) = f_2(2)$

    (yes, technically, $\displaystyle x = 2$ is not in the domain of $\displaystyle f_2(x)$, but we do this because we want to get arbitrary close to $\displaystyle x = 2$)
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  3. #3
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    Given $\displaystyle F(x) = \left\{ {\begin{array}{c}
    {5x - 4,\;x \le 2} \\
    { - 4x + b,\;x > 2} \\
    \end{array}} \right.$

    You want: $\displaystyle \lim _{x \to 2^ - } F(x) = \lim _{x \to 2^ + } F(x)$
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