I am somewhat confused as to what you are asking. I assume you want to show that is a subset of the Borel sigma algebra of the domain space.
Take any , then there exists a B in the Borel sigma algebra of the range, such that . As B is a Borel set, it is the intersection of some open sets (countable union of open sets is also open, all we are missing is the countable intersection), so . Now , and as f is cts, each is open in the domain. Hence A is a Borel set.