Question

Discuss the continuity of the function $f\left(x\right)=\left\{\begin{array}{ll}\frac{x}{\left|x\right|},& x\ne 0\\ 0,& x=0\end{array}.\right.$

Answer 1

Given: $f\left(x\right)=\left\{\begin{array}{l}\frac{x}{\left|x\right|},x\ne 0\\ 0,x=0\end{array}\right.$

$$\begin{gathered} \left|x\right|=\left\{\begin{array}{l}x,x\ge 0\\ -x,x<0\end{array}\right. \\ \Rightarrow f\left(x\right)=\left\{\begin{array}{c}1,x>0\\ -1,x<0\\ 0,x=0\end{array}\right. \end{gathered}$$

We have
(LHL at x = 0) = $\underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(0-h\right)=\underset{h\to 0}{\lim }f\left(-h\right)=\underset{h\to 0}{\lim }\left(-1\right)=-1$

(RHL at x = 0) = $\underset{x\to 0^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(0+h\right)=\underset{h\to 0}{\lim }f\left(h\right)=\underset{h\to 0}{\lim }\left(1\right)=1$

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

Thus, $f\left(x\right)$ is discontinuous at x = 0.