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This might be a simple question, but it is really confusing. I have not found any explicit definition or example of extended real-valued function.

My guess is that f defined by f: x -> 1/x on (0,1) is REAl-valued, while g defined by g: x -> 1/x on [0,1) is EXTENDED REAL-valued. Is this the case?

Could someone please give some examples?

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    $\begingroup$ It just means takes on a symbolic value called infinity (and minus infinity). $\endgroup$
    – IAmNoOne
    Sep 23, 2015 at 3:43
  • $\begingroup$ @Hong-BinChen Concrete example, as you ask on the comment below the answer, are simply making a function attain these values. $f \equiv +\infty$ is an example. If you want some more information (specifically, topological information), I can indicate you to this blog post I made on the subject. If you want some information regarding measure theory (where the extended real line is very useful), you can consult any Measure Theory book. $\endgroup$
    – Aloizio Macedo
    Sep 23, 2015 at 3:51

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In general, an $E$-valued function is a function that takes values in $E$.

The extended reals are $\mathbb{R}\cup\{\pm \infty\}$. So an extended-reals-valued function takes values that are either in $\mathbb{R}$, or possibly $\pm \infty$.

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  • $\begingroup$ Thank you! But could you please show me some concrete examples. I know the definition, and want to have a touch on some simple but tangible examples. $\endgroup$
    – user256172
    Sep 23, 2015 at 3:45
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    $\begingroup$ @Hong-BinChen For instance, you might try to show that $1/x^2$ is a continuous function to the extended reals (with the natural topology that you think should happen, which also happens to be the 2-point compactification). $\endgroup$
    – davidlowryduda
    Sep 23, 2015 at 3:52
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The only difference between Real valued function and Extended Real valued function is interval of domain . . . In case of Real Valued Function Domain is (-∞,+∞) and in case of Extended Real Valued Function Domain is [-∞,+∞]. Now another Question arises that ∞ is a real number? absolutely no . . . But in mathematical analysis and calculus ∞ play an important role that's why it is named as extended real numbers , moreover it also satisfies some laws of binary operation (i.e addition and multiplication) just as follows 5 + ∞ = ∞ + 5 = ∞ 5 * ∞ = ∞ * ∞ = ∞

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