continuous functions on the extended real numbers

Within this entry, $\overline{\mathbb{R}}$ will be used to refer to the extended real numbers.

Theorem 1.

Let $f\colon\mathbb{R}\to\mathbb{R}$ be a function. Then $\overline{f}\colon\overline{\mathbb{R}}\to\overline{\mathbb{R}}$ defined by

$\overline{f}(x)=\begin{cases}f(x)&\text{ if }x\in\mathbb{R}\\ A&\text{ if }x=\infty\\ B&\text{ if }x=-\infty\end{cases}$

is continuous if and only if $f$ is continuous such that $\displaystyle\lim_{x\to\infty}f(x)=A$ and $\displaystyle\lim_{x\to-\infty}f(x)=B$ for some $A,B\in\overline{\mathbb{R}}$ .

Proof.

Note that $\overline{f}$ is continuous if and only if $\displaystyle\lim_{x\to c}\overline{f}(x)=\overline{f}(c)$ for all $c\in\overline{\mathbb{R}}$ . By defintion of $\overline{f}$ and the topology of $\overline{\mathbb{R}}$ , $\displaystyle\lim_{x\to c}\overline{f}(x)=\displaystyle\lim_{x\to c}f(x)$ for all $c\in\overline{\mathbb{R}}$ . Thus, $\overline{f}$ is continuous if and only if $\displaystyle\lim_{x\to c}f(x)=\overline{f}(c)$ for all $c\in\overline{\mathbb{R}}$ . The latter condition is equivalent (http://planetmath.org/Equivalent3) to the hypotheses that $f$ is continuous on $\mathbb{R}$ , $\displaystyle\lim_{x\to\infty}f(x)=A$ , and $\displaystyle\lim_{x\to-\infty}f(x)=B$ . ∎

Note that, without the universal assumption that $f$ is a function from $\mathbb{R}$ to $\mathbb{R}$ , necessity holds, but sufficiency does not. As a counterexample to sufficiency, consider the function $\overline{f}\colon\mathbb{R}\to\mathbb{R}$ defined by

$\overline{f}(x)=\begin{cases}\displaystyle\frac{1}{x^{2}}&\text{ if }x\in% \mathbb{R}\setminus\{0\}\\ \infty&\text{ if }x=0\\ 0&\text{ if }x=\pm\infty.\end{cases}$

Title	continuous functions on the extended real numbers
Canonical name	ContinuousFunctionsOnTheExtendedRealNumbers
Date of creation	2013-03-22 16:59:31
Last modified on	2013-03-22 16:59:31
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	10
Author	Wkbj79 (1863)
Entry type	Theorem
Classification	msc 12D99
Classification	msc 28-00