continuous functions on the extended real numbers

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

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

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 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