I have a problem with the two last questions of this exercise.
1) At which points is the absolute value function f(x) = |x| continuous?
$\forall x \in \mathbb{R}$ f is continuous
2) What about $f(x)=\frac{|x|}{x}$
$\forall x \in \mathbb{R}-{0}$, f is continuous
3) And what about $f(x)=\begin{cases} \frac{|x|}{x} &\text{if }x \neq 0\\0 &\text{if }x=0\end{cases}$
4) What type of discontinuities appear if any at all?
Thank you in advance