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  1. MHB Master
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    #1
    I am reading Cesar E. Silva's book entitled "Invitation to Real Analysis" ... and am focused on Chapter 4: Continuous Functions ...

    I need help to clarify an aspect of the proof of Theorem 4.2.1, the Intermediate Value Theorem ... ...

    Theorem 4.2.1 and its related Corollary read as follows:








    In the above proof by Silva, we read the following:

    " ... ... So there exists $ \displaystyle x$ with $ \displaystyle b \gt x \gt \beta$ and such that $ \displaystyle f(x) \lt 0$ ... ... "


    My question is as follows:

    How can we be sure that $ \displaystyle f(x) \lt 0$ given $ \displaystyle x$ with $ \displaystyle b \gt x \gt \beta$ ... indeed how do we show rigorously that for $ \displaystyle x$ such that $ \displaystyle b \gt x \gt \beta$ we have $ \displaystyle f(x) \lt 0$ ...


    Help will be much appreciated ...

    Peter

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  3. MHB Craftsman
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    #2
    Hi Peter,

    Since $\beta<b$ and $\delta >0$, there are values of $x$ such that $|x-\beta|<\delta$ and $\beta < x< b$. The key here is to remember that $|x-\beta| <\delta$. By the continuity argument, $f(x)<0$ for all such $x$.

  4. MHB Master
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    Quote Originally Posted by GJA View Post
    Hi Peter,

    Since $\beta<b$ and $\delta >0$, there are values of $x$ such that $|x-\beta|<\delta$ and $\beta < x< b$. The key here is to remember that $|x-\beta| <\delta$. By the continuity argument, $f(x)<0$ for all such $x$.





    Thanks for the help GJA!

    At first I struggled with what you meant by ... " By the continuity argument, $f(x)<0$ for all such $x$ ... "

    But then I found Apostol Theorem 3.7 (Calculus Vol. 1, page 143) which reads as follows:





    Were you indeed invoking something like what Apostol calls the sign-preserving property of continuous functions?


    Thanks again for your help ...

    Peter

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    #4
    Hi Peter,

    Happy to help!

    I wasn't quoting that purposely, though it is true. In fact, it's essentially what the author is proving by their choice of epsilon.

    What I meant was: $|f(x)-f(\beta)|<\epsilon\,\Longrightarrow\, f(x)<\epsilon + f(\beta)<0.$

    Hope this helps clear up the confusion on my earlier post.

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