That is not true unless your function is continuous. For example:
Let define the function:
. But this is the preimage of the open interval (1,2) whose preimage is neither open nor closed.
The question's listed below:
Let f be a real-valued function defined on R (real). Show that the inverse image with respect to f of an open set is open, of a closed set is closed, and of a Borel set is Borel.
Thank you so much for the help!