continuous functions on the extended real numbers

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

Theorem 1.

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

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

Proof.

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

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