Answer
See proof
Work Step by Step
We are given:
$f(x)=|x|$
Rewrite the function:
$f(x)=\begin{cases}
-x,\text{ for }x\leq 0\\
x,\text{ for }x>0
\end{cases}$
Compute the left hand and right hand limits:
$\displaystyle\lim_{x\rightarrow 0^{-}} f(x)=\displaystyle\lim_{x\rightarrow 0^{-}} (-x)=-0=0$
$\displaystyle\lim_{x\rightarrow 0^{+}} f(x)=\displaystyle\lim_{x\rightarrow 0^{+}} x=0$
As $\displaystyle\lim_{x\rightarrow 0^{-}} f(x)=\displaystyle\lim_{x\rightarrow 0^{+}} f(x)=f(0)=0$,
the function is continuous in $x=0$. The function is also continuous on $(-\infty,0)$ and $(0,\infty)$,therefore it is continuous on $\mathbb{R}$.
