TITLE: Does there exist a function that is continuous at every rational point and discontinuous at every irrational point? And vice versa?
QUESTION [6 upvotes]: Actually there are 2 questions, but they are closely related.
Does it exist a function that is:  

Continuous  at every rational point and discontinuous at every irrational point?
Continuous  at every irrational point and discontinuous at every rational point?

REPLY [2 votes]: For first part there does not exist  any function
Because f is discontinuous on only F-sigma set. F-sigma set is defined as the set which can be written as countable union of closed set.