Question

Find $\underset{x\to 0}{lim}f\left(x\right)$ where $f\left(x\right)=\{\begin{array}{ccc}x-1& if& x<0\\ 0& if& x=0\\ x+1& if& x>0\end{array}$

Answer 1

$f\left(x\right)=\{\begin{array}{c}x-1;x<0\\ 0;x=0\\ x+1;x>0\end{array}$

Checking continuity for $x=0$

$f$ is continuous at $x=0$ if LHL$=$RHL$=f\left(0\right)$

${lim}_{x\to 0^{-}}f\left(x\right)={lim}_{x\to 0^{+}}f\left(x\right)=f\left(0\right)$

LHL$={lim}_{x\to 0^{-}}f\left(x\right)={lim}_{x\to 0^{-}}(x-1)=0-1=-1$ as $x<0$

RHL$={lim}_{x\to 0^{+}}f\left(x\right)={lim}_{x\to 0^{+}}(x+1)=0+1=1$ as $x>0$

Since LHL$\ne$RHL

Hence $f$ is not continuous at $x=0$

$\therefore f$ is continous for $x\in R-\left\{0\right\}$