Question

Discuss the continuity of the function $f\left(x\right)$ at $x=0$.
$f\left(x\right)=\{\begin{array}{c}\frac{x-|x|}{x};x\ne 0\\ x=2;x<0\end{array}$

Answer 1

$f\left(x\right)=\{\begin{array}{c}1-\frac{\left|x\right|}{x};x\ne 0\\ 2;x=0\end{array}$

${lim}_{x\to 0^{+}}f\left(x\right)={lim}_{x\to 0^{+}}(1-\frac{\left|x\right|}{x})=1-1=0$

as $x\to 0^{+}\Rightarrow x>0\Rightarrow \frac{\left|x\right|}{x}=1$

${lim}_{x\to 0^{+}}f\left(x\right)=1+1=2$

as $x\to 0^{-}\Rightarrow x<0\Rightarrow \frac{\left|x\right|}{x}=-1$

RHL$\ne$LHL $\Rightarrow$Discontinuous at $x=0$