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Math Help - Functions Where the Extended Real Line gives (or does not give) Continuity

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    [SOLVED] Functions Where the Extended Real Line gives (or does not give) Continuity

    I'm examining the final paragraph on page 75 from, Oden - Applied Functional Analysis 2nd ed., and am hoping for clarity.

    A statement is made that for a function such as \frac{1}{x^2} using the extended real line where the value at x = 0 is \infty, is continuous. The author then goes on to say that this continuity is not true for \frac{1}{x} at x=0.

    I am looking for illumination as to why this is the case. Thank you.

    I am not sure if this may be of help but the section is "Functions with values in Bar-R (Extended Real Line)." Previous discussions include point-wise sup/inf.

    Chris
    Last edited by ZircX; February 1st 2011 at 07:26 AM. Reason: Solved
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    As x approaches zero from either side, 1/x^2 goes to + infinity. As x approaches zero from either side, 1/x goes to either +infinity or -infinity. The extended reals include both infinities, and they are not equal. So you get a jump discontinuity at 0.
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    You're welcome!
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  5. #5
    MHF Contributor FernandoRevilla's Avatar
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    Another thing would be if we compatify \mathbb{R} with only one point i.e. \overline{\mathbb{R}}=\mathbb{R}\cup \{\infty\}. In this case, \lim_{x\to 0} (1/x)=\infty .


    Fernando Revilla
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