Question

Prove that $f\left(x\right)=\left\{\begin{array}{ll}\frac{x-\left|x\right|}{x},& x\ne 0\\ 2,& x=0\end{array}\right.$is discontinuous at x = 0

Answer 1

The given function can be rewritten as

$f\left(x\right)=\left\{\begin{array}{c}\frac{x-x}{x},\mathrm{when}x>0\\ \frac{x+x}{x},\mathrm{when}x<0\\ 2,\mathrm{when}x=0\end{array}\right.$

​$\Rightarrow$ $f\left(x\right)=\left\{\begin{array}{c}0,\mathrm{when}x>0\\ 2,\mathrm{when}x<0\\ 2,\mathrm{when}x=0\end{array}\right.$

We have
(LHL at x = 0) = $\underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(0-h\right)=\underset{h\to 0}{\lim }f\left(-h\right)=\underset{h\to 0}{\lim }2=2$

(RHL at x = 0) = $\underset{x\to 0^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(0+h\right)=\underset{h\to 0}{\lim }f\left(h\right)=\underset{h\to 0}{\lim }0=0$


∴ ​$\underset{x\to 0^{-}}{\lim }f\left(x\right)\ne \underset{x\to 0^{+}}{\lim }f\left(x\right)$

Thus, f(x) is discontinuous at x = 0.