Lower limit topology with continuous function

Question

I want to prove that the only continuous function from $R$ (with usual topology) to $R$ ( the Sorgenfrey line's $R_l$ Lower limit topology) is constant function.

My attempt is using the the properties of connected space. BOWC, assume there is continuous function which are not constant and surjective Since $f(R)$ is connected with more than one points. Then, it must be interval and $f(R)$ as subspace of $R_l$. I can not complete. I am asking wether or not my solution right.

Answer 1

HINT: You need to show that if $Y$ is a subspace of $\Bbb R_\ell$ with more than one point, then $Y$ is not connected. To do this, let $Y$ be such a subspace, and let $x,y\in Y$ with $x<y$. Let

$$L=\{z\in Y:z<y\}$$

and

$$R=\{z\in Y:z\ge y\}\;.$$

Show that $Y=L\cup R$, $L\cap R=\varnothing$, and $L$ and $R$ are both open in the subspace $Y$.