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

1. ## [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 $\displaystyle \frac{1}{x^2}$ using the extended real line where the value at x = 0 is $\displaystyle \infty$, is continuous. The author then goes on to say that this continuity is not true for $\displaystyle \frac{1}{x}$ 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

2. 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.

3. Thanks

4. You're welcome!

5. Another thing would be if we compatify $\displaystyle \mathbb{R}$ with only one point i.e. $\displaystyle \overline{\mathbb{R}}=\mathbb{R}\cup \{\infty\}$. In this case, $\displaystyle \lim_{x\to 0} (1/x)=\infty$ .

Fernando Revilla