You can't expect to just tell you the answer, right? We need to see some work? What is the Sorgenfrey topology? Of course, it is what usually referred to as the

Sorgenfrey line or "lower limit topology", the topology which has \(\displaystyle \mathfrak{B}=\left\{[a,b):a,b\in\mathbb{R}\right\}\) as a base?

So, what are your ideas? You should know that if it's continuous it is sufficient to show that the preimage of every basic open set is open. So, what are the cases for \(\displaystyle f^{-1}\left([a,b]\right)\)?