examples of continuous functions on the extended real numbers

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

Examples of continuous functions on $\overline{\mathbb{R}}$ include:

•

Polynomial functions: Let $f\in\mathbb{R}[x]$ with $\displaystyle f(x)=\sum_{j=0}^{n}a_{n}x^{n}$ for some $n\in\mathbb{N}$ and $a_{0},\ldots,a_{n}\in\mathbb{R}$ with $a_{n}\neq 0$ if $n\neq 0$ . Then $\overline{f}$ is defined in the following manner:
1. (a)
  
  If $n=0$ , then $\overline{f}(x)=a_{0}$ for all $x\in\overline{\mathbb{R}}$ .
2. (b)
  
  If $n$ is odd and $a_{n}>0$ , then $\displaystyle\overline{f}(x)=\begin{cases}f(x)&\text{ if }x\in\mathbb{R}\\ x&\text{ if }x\notin\mathbb{R}.\end{cases}$
3. (c)
  
  If $n$ is odd and $a_{n}<0$ , then $\displaystyle\overline{f}(x)=\begin{cases}f(x)&\text{ if }x\in\mathbb{R}\\ -x&\text{ if }x\notin\mathbb{R}.\end{cases}$
4. (d)
  
  If $n\neq 0$ is even and $a_{n}>0$ , then $\displaystyle\overline{f}(x)=\begin{cases}f(x)&\text{ if }x\in\mathbb{R}\\ \infty&\text{ if }x\notin\mathbb{R}.\end{cases}$
5. (e)
  
  If $n\neq 0$ is even and $a_{n}<0$ , then $\displaystyle\overline{f}(x)=\begin{cases}f(x)&\text{ if }x\in\mathbb{R}\\ -\infty&\text{ if }x\notin\mathbb{R}.\end{cases}$
•

Exponential functions: Let $f(x)=a^{x}$ for some $a\in\mathbb{R}$ with $a>0$ and $a\neq 1$ . Then $\overline{f}$ is defined in the following manner:
1. (a)
  
  If $a<1$ , then $\displaystyle\overline{f}(x)=\begin{cases}f(x)&\text{ if }x\in\mathbb{R}\\ 0&\text{ if }x=\infty\\ \infty&\text{ if }x=-\infty.\end{cases}$
2. (b)
  
  If $a>1$ , then $\displaystyle\overline{f}(x)=\begin{cases}f(x)&\text{ if }x\in\mathbb{R}\\ \infty&\text{ if }x=\infty\\ 0&\text{ if }x=-\infty.\end{cases}$
•

Miscellaneous
1. (a)
  
  Let $f(x)=\arctan x$ . Then $\overline{f}$ is defined by $\displaystyle\overline{f}(x)=\begin{cases}f(x)&\text{ if }x\in\mathbb{R}\\ &\\ \displaystyle\frac{\pi}{2}&\text{ if }x=\infty\\ &\\ \displaystyle-\frac{\pi}{2}&\text{ if }x=-\infty.\end{cases}$
2. (b)
  
  Let $f(x)=\tanh x$ . Then $\overline{f}$ is defined by $\displaystyle\overline{f}(x)=\begin{cases}f(x)&\text{ if }x\in\mathbb{R}\\ 1&\text{ if }x=\infty\\ -1&\text{ if }x=-\infty.\end{cases}$

Of course, not every function $f$ that is continuous on $\mathbb{R}$ extends to a continuous function on $\overline{\mathbb{R}}$ . Common examples of these include the real functions $x\mapsto\sin x$ and $x\mapsto\cos x$ . (It is proven that these are continuous on $\mathbb{R}$ in the entry continuity of sine and cosine.)

On the other hand, there are some continuous functions $\overline{f}\colon\overline{\mathbb{R}}\to\overline{\mathbb{R}}$ that have no analogous function $f\colon\mathbb{R}\to\mathbb{R}$ . For example, consider

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

Title	examples of continuous functions on the extended real numbers
Canonical name	ExamplesOfContinuousFunctionsOnTheExtendedRealNumbers
Date of creation	2013-03-22 16:59:34
Last modified on	2013-03-22 16:59:34
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	9
Author	Wkbj79 (1863)
Entry type	Example
Classification	msc 12D99
Classification	msc 28-00