The function $f$ is defined in $[-5,5]$ as
$
f(x)=
\begin{cases}
x, & \text{if }x\text{ is rational}\\
-x, & \text{if }x\text{ is irrational}
\end{cases}
$
For rationals $f(x)=x$, for irrationals $f(x)=-x$.
At $x=0$, both give $0$, so it’s continuous there.
At any $x\neq0$, limits from rationals and irrationals differ.