$\epsilon$-$\delta$ 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.

Hint: Check if the sequence $f(1+\frac{\sqrt{2}}{n})$ becomes very close to $f(1)$ for large $n$
To do it using $\epsilon,\delta$ definition. Suppose there exists $\delta>0$ such that for $\forall y\in \mathbb{R}[|x-y|<\delta\implies|f(x)-f(y)|<0.1]$. Now choose $n$ sufficently large such that $|(1+\frac{\sqrt{2}}{n})-1|<\delta$ (Use the archemidean principle to do this). Thus $|f(1+\frac{\sqrt{2}}{n})-f(1)|<0.1$ Does this lead to a contradiction ?
