In an intersection between a regular language and a DCFL, we always tend to promote regular language to DCFL and say that the result will be intersection between DCFL and DCFL but since DCFLs are not closed under intersection we say the result will be a CFL. But I feel the answer to the original question should be DCFL because I never came across a language which is an intersection between a Regular language and a DCFL which is not DCFL
Can you write a language which is an intersection between a DCFL and a regular language but not a DCFL.