I have a question based on the epsilon delta definition of continuity.
f(x) = 1, when x is rational and 0 when x is irrational.
Prove that f is discontinuous at every point.
Yes, that is a fundamental property of the rational and irrational numbers- they are both "dense" in the real numbers. In any interval of real numbers, no matter how small, there exist both rational and irrational numbers. Whoever gave you this problem expected you to know that.