For what value(s) of x is f continuous?
f(x)= { 0 if x is rational
2 if x is irrational
I am going to use the sequential definition of continuity and show the definition fails at rational and irrational points. It is not continuous anywhere. Let $\displaystyle a$ be rational. Let $\displaystyle x_{n} = \frac{1}{n\sqrt{2}} + a$, which is a sequence of irrational numbers which converges to $\displaystyle a$. Since each $\displaystyle x_{n}$ is irrational, $\displaystyle f(x_{n}) = 2$ for all positive integers $\displaystyle n$, so it trivially converges to $\displaystyle 2$, and since $\displaystyle f(a) = 0 \ne 2$, we have that $\displaystyle \lim x_{n} = a$ yet $\displaystyle \lim f(x_{n}) \ne f(a)$, so the function is discontinuous on the rationals. Now let $\displaystyle a$ be irrational. By the denseness of $\displaystyle \mathbb{Q}$, we can construct a sequence of rational numbers $\displaystyle x_{n}$ that converges to $\displaystyle a$. Since each $\displaystyle x_{n}$ is rational, we have that $\displaystyle f(x_{n}) = 0$ for all positive integers $\displaystyle n$. So $\displaystyle f(x_{n})$ trivially converges to $\displaystyle 0$, and since $\displaystyle f(a) = 2 \ne 0$, we have that $\displaystyle \lim x_{n} = a$ yet $\displaystyle \lim f(x_{n}) \ne f(a)$, so the function is discontinuous on the irrationals.
If you need to find the values using the epsilon-delta definition of continuity, the basic idea behind the proof is that any interval contains infinitely many rationals and irrationals, so if the function were continuous then $\displaystyle f(x)$ must get arbitrarily close to 0 and 2 simultaneously, which is clearly impossible.