I was reading Dirichlet and Thomae's functions and got interested to know about functions which are continuous nowhere. Since these have a lot to do with rationals and irrationals, the next question that came to my mind is:
what about a function which is rational at all irrational points and irrational at all rational points?
Is this function continuous?
I will tell you a brief sketch of my approach. I first proved that if our function assumes a constant rational value for every irrational, but varying irrational values for the rationals, then by Sequential Criterion of Continuity, this function is not continuous. But what about the general case i.e. the function assumes varying irrational values for varying rationals and varying rational values for varying irrationals? I could not proceed further.
Any hint or help will be appreciated.