A). (I'm not that good at these so it would be dandy if someone could check this over).
Let C be our rational number. Hence .
Let's construct a sequence such that and a sequence (where ).
Clearly, approaches C from the right and approaches C from the left. Our goal is to show that and are different.
is always rational so is always rational. Hence .
is always irrational so is always irrational. Hence .
so the function f is not continuous at every rational number.