Problem: Test the following function for continuity. $$f(x)=\begin{cases} x^2 & x\text{ is rational, } \\ -x^2 & x\text{ is irrational. } \\ \end{cases}$$
My attempt: Pick $x_0\in \mathbb{R}$. Then for any $\varepsilon>0$ we have to find $\delta>0$ such that at $|x-x_0|<\delta$, the inequality $|f(x)-f(x_0)|<\varepsilon$ holds. I suspect that this function is discontinuous everywhere and in order to prove that one usually guesses a value for $\varepsilon$ such that inequality $|f(x)-f(x_0)|<\varepsilon$ never holds. What value should I pick for $\epsilon$ ?