Talk:Extended real number line

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[edit] notation

I question the notation in this article. should the symbol for the extended reals be R? that's the same symbol for the regular real line. using the same symbol for both could be endlessly confusing. None of my textbooks (Rudin, Royden, Schechter) uses this notation, but they also don't seem to have a prefered notation. They just call it by name. I think this article should follow that same convention. -Lethe | Talk

-- It isn't R, but R with a bar on top.

Which displays for some people, including me, as just R. Dbenbenn 00:02, 25 Dec 2004 (UTC)

[edit] Failure of distributive property

It is currently stated that "a × (b + c) and (a × b) + (a × c) are either equal or both undefined." but unfortunately this is incorrect. Consider a = +oo, b = 2 and c = -1, for example. Then a × (b + c) is defined, but (a × b) + (a × c) is not.

I'm not sure how the entry should best be corrected.

Good catch! I think the following modification is true
  • a × (b + c) and (a × b) + (a × c) are equal if both are defined.
Dbenbenn 02:15, 29 Dec 2004 (UTC)

[edit] not the example I want

f_n(x) = \begin{cases} 2n(1-nx), & \mbox{if } 0 \le x \le \frac{1}{n} \\ 0, & \mbox{if } \frac{1}{n} < x \le 1\end{cases}

This is not the example I want. This is an example to show that you can't interchange the limits at will. I want an example to show that it's OKAY to get the limit function to take on ∞ at one point say, and still satisfy one of the convergence theorems. I can't come up with the right example at the moment, maybe my brain is fried. If anyone remembers an example like this, feel free to put it there. It's one of those standard examples, I just can't think of it. Revolver 19:31, 16 September 2005 (UTC)

[edit] Move to "Extended real number system"?

The line is just a way of visualizing the set in question. IMHO "Extended real number system" would be a more suitable title. --Kprateek88(Talk | Contribs) 06:27, 21 November 2006 (UTC)

Or better, "affinely extended real number system", as opposed to the projectively extended real number system. -- Schapel 07:47, 21 November 2006 (UTC)

[edit] "the function y = x − 2 can be made continuous by setting the value to +∞ for x = 0"

This article seems to play fast and loose with the definition of "continuous". Under the normal definition of continuity, small changes in x should produce relatively small changes in y. This is clearly not the case where x=0, where changes in y become unbounded. The function is continuous by the topological definition, but this is not clear cojoco 03:47, 6 November 2007 (UTC)

Which other definition is there? FilipeS (talk) 02:38, 30 November 2007 (UTC)
There are at least two definitions: under the first definition in Continuous_function, this function is not continuous, because nearby differences become unbounded at x=0. Under topological definition in the same article, it is. The statement "the function y = x − 2 can be made continuous by setting the value to +∞ for x = 0" is misleading because this is not true under the normal definition of continuity, so the statement should be clarified. So, I've clarified it.cojoco (talk) 21:21, 21 December 2007 (UTC)
You are right, in so far as in the standard definition of continuity, when we say that "the limit of f(x) as x approaches c must exist and be equal to f(c)", we assume that f is a real-valued function. Thus, f(c) cannot be infinite, by definition. FilipeS (talk) 17:12, 11 January 2008 (UTC)