$\epsilon$-$\delta$ proof of discontinuity How can I prove that the function defined by
$$f(x) = \begin{cases} x^{2}, & \text{if $x \in \mathbb{Q}$;} \\ -x^{2}, & \text{if  $x \notin \mathbb{Q}$;} \end{cases} $$ is discontinuous? 
I see that it is true by using sequences but I cannot prove using only $\epsilon$'s and $\delta$'s.

Lets show that $f$ is discontinuous at $a>0$.
For each $\delta>0$ we can find $x_1,x_2\in (a-\delta,a+\delta), \ x_1\in\mathbb Q, \ x_2\not\in\mathbb Q$ such that $f(x_1)>\dfrac{a^2}{2}, \ f(x_2)<-\dfrac{a^2}{2}$. Therefore $\forall \delta>0 \ \exists x\in(a-\delta,a+\delta)$ such that $|f(x)-f(a)|>\dfrac{3a^2}{2}\Longrightarrow f$ is discontinuous at $a$.
