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 .

Therefore:

so the function f is not continuous at every rational number.