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

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