Suppose $A\subseteq\mathbb{R}_l$ is connected and $x\in A$ . $[x,x+\epsilon)\cap A$ is both open and closed in $A$ and nonempty. Therefore, using §23 criterion, $A=A\cap[x,x+\epsilon)$ for any $\epsilon$ and $A=\{x\}$ . So, each component of $\mathbb{R}_l$ is a singleton. Hence, path components are also singletons. Now, the image of a connected space under a continuous mapping must be connected. Therefore, only constant functions are continuous functions from $\mathbb{R}$ to $\mathbb{R}_l$ . Compare to §18, exercise 7(b).