Limit of a piecewise function defined by x being rational or irrational

Question

If

$$ f(x)=\begin{cases} x² \quad \quad \text { if x is rational} \newline 0 \ \ \ \ \ \ \ \ \ \ \text {if x is irrational} \end{cases} $$

prove that $\lim_{x \to 0}f(x) = 0$

Answer 1

You can use the squeeze theorem and the fact that $-x^2 \le f(x) \le x^2$