Let a function $f$ be defined by $f\left(x\right)=\frac{x-\left|x\right|}{x}\text{ for }x\ne 0$ and $f\left(0\right)=2$ then $f$ is

$f\left(x\right)=\left\{\begin{array}{l}2,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if }x<0\\ 0,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if }x>0\\ 2,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if }x=0\end{array}\right.$

$$\begin{gathered} \text{ Thus }\underset{x\to 0}{lim} f\left(x\right)=2\ne 0=\underset{x\to 0^{+}}{lim} f\left(x\right) \\ \Rightarrow f is not continues at x=0 \end{gathered}$$

Hence, $f$is continuous every where except at $x = 0$.