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

**ZircX** · February 1st 2011, 07:17 AM

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

**Ackbeet** · February 1st 2011, 07:23 AM

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.

**ZircX** · February 1st 2011, 07:26 AM

Thanks

**Ackbeet** · February 1st 2011, 07:28 AM

You're welcome!

**FernandoRevilla** · February 1st 2011, 07:30 AM

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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