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

$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}2,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}<0\\ 0,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}>0\\ 2,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if }\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}=0\end{array}\right.$Thus $\underset{\mathrm{x}\to 0^{-}}{lim} \mathrm{f}\left(\mathrm{x}\right)=2\ne 0=\underset{\mathrm{x}\to 0^{+}}{lim} \mathrm{f}\left(\mathrm{x}\right)$

f(x) is discontinuous at x=0
Hence, f is continuous every where except at x = 0.