# Real-valued function and extended-real-valued function, what's the difference?

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?

• It just means takes on a symbolic value called infinity (and minus infinity). Sep 23, 2015 at 3:43
• @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. Sep 23, 2015 at 3:51

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 $\pm \infty$.
• @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). Sep 23, 2015 at 3:52