## Calculus (3rd Edition)

We are given the function: $f(x)=\begin{cases} x,\text{ for }x \text{ rational}\\ -x,\text{ for }x \text{ irrational} \end{cases}$ First we proved that $f$ is not continuous at $x_0\not=0$. We take a sequence of rational numbers $x_n$ converging to $x_0$ and a sequence of irrational numbers $y_n$ converging to $x_0$. We see that $\displaystyle\lim_{x\rightarrow x_0} f(x)$ does not exist. Then we prove that $f(x)$ is continuous in $x=0$. $f(0)=0$ $-|x|\leq f(x)\leq |x|$ for all $x$. $\displaystyle\lim_{x\rightarrow 0} (-|x|)=\displaystyle\lim_{x\rightarrow 0} |x|=0$ From the pinching theorem we get: $\Rightarrow \displaystyle\lim_{x\rightarrow 0} f(x)=0$ This means that $f$ is continuous only in $x=0$.