examples of continuous functions on the extended real numbers

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

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

• Polynomial functions: Let $f\in ℝ[x]$ with $f(x)={\displaystyle \sum _{j=0}^n}{a}_n{x}^n$ for some $n\in \mathbb{N}$ and $a_0,\mathrm{\dots },a_n\in ℝ$ with $a_n\ne 0$ if $n\ne 0$. Then $\overline{f}$ is defined in the following manner:

  1. If $n=0$, then $\overline{f}(x)=a_0$ for all $x\in \overline{ℝ}$.

  2. If $n$ is odd and $a_n>0$, then

  3. If $n$ is odd and , then

  4. If $n\ne 0$ is even and $a_n>0$, then

  5. If $n\ne 0$ is even and , then

• Exponential functions: Let $f(x)=a^x$ for some $a\in ℝ$ with $a>0$ and $a\ne 1$. Then $\overline{f}$ is defined in the following manner:

  1. If , then

  2. If $a>1$, then

• Miscellaneous

  1. Let $f(x)=\arctan x$. Then $\overline{f}$ is defined by

  2. Let $f(x)=\tanh x$. Then $\overline{f}$ is defined by

Of course, not every function $f$ that is continuous on $ℝ$ extends to a continuous function on $\overline{ℝ}$. 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 $ℝ$ in the entry continuity of sine and cosine.)

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