Consider $\mathbb{R}_l$ be the lower limit topology which consists basis element of the form $[a,b)$ where $a<b$ and $a,b \in \mathbb{R}$.
I am facing trouble to solve what are the continuous functions from $\mathbb{R}$ to $\mathbb{R}_l$.
Hint: Only constant functions are continuous from $\mathbb{R}$ to $\mathbb{R}_l$. Indeed, the inverse image of any $[a,b)$ has to be both open and closed, and there are only two such sets in $\mathbb{R}$ , namely, the empty set and $\mathbb{R}$.
But, I have no idea how to solve it? Please help me.
Let $f:\mathbb R\to\mathbb R_\ell$ be continuous. Let $c=f(0).$ You should try to show that $f$ is just the constant map $x\mapsto c.$ To do this you should use the hint. Namely take an interval $[a,b)$ and show that its preimage is either empty or all of $\mathbb R$ depending on whether it contains $c$ or not. Can you do it from here?