Can one use these facts to show that the Thomae function (aka the popcorn function) $ t(x) = \begin{cases} 0 & \text{if}~ x~\text{is irrational}\\ 1/n & \text{if}~x=m/n,~\text{where}~m,n\in \mathbb N,n\gt 0,\gcd(m,n) =1 \end{cases} $
is continuous at every irrational points and discontinuous at all rational points in $\mathbb R$ ?
Facts:
(i) the set of all irrational numbers in $\mathbb R$ is not an $F_{\sigma}$ set.
(ii) the set of points of continuity of any function $f$ is a $G_{\delta}$ set.
(iii) there is no function that is discontinuous at all irrational points and continuous at all rational points .