The function f(x) is defined by $f\left(x\right)=\{\begin{array}{c}\frac{\left|x\right|}{x}\\ 0,\end{array}\begin{array}{c},x\ne 0\\ x=0\end{array}$ then, at x = 0 it is

Concept:

f(x) is continuous at x = a, if LHL = RHL = f(a)

 $\underset{\mathrm{x}\to {\mathrm{a}}^{-}}{lim}\mathrm{f}(\mathrm{x})=\underset{\mathrm{x}\to {\mathrm{a}}^{+}}{lim}\mathrm{f}(\mathrm{x})=\mathrm{l}\mathrm{i}{\mathrm{m}}_{\mathrm{x}\to \mathrm{a}}\mathrm{f}(\mathrm{x})$

 f(x) is differentiable if LHD = RHD

$\begin{array}{l}\mathrm{L}\mathrm{H}\mathrm{D}=\underset{\mathrm{h}\to 0^{-}}{lim}\frac{\mathrm{f}(\mathrm{a}-\mathrm{h})-\mathrm{f}(\mathrm{a})}{-\mathrm{h}}\\ \mathrm{R}\mathrm{H}\mathrm{D}=\underset{\mathrm{h}\to 0^{+}}{lim}\frac{\mathrm{f}(\mathrm{a}+\mathrm{h})-\mathrm{f}(\mathrm{a})}{\mathrm{h}}\end{array}$

Discontinuity of the First Kind: A function f(x) is said to have a discontinuity of the first kind from the right at x = a if the right hand of the function exists but not equal to f(a).

Discontinuity of the Second Kind: A function f(x) is said to have discontinuity of the second kind at x = a, if neither left-hand limit of f(x) at x = a nor right-hand limit of f(x) at x = a exists.

Removable Discontinuity: A function f(x) is said to have a removable discontinuity at x = a if the left-hand limit at x tends to point ‘a’ is equal to the right-hand limit at x tends to point ‘a’ but their common value is not equal to f(a). 

Calculation:

$f\left(x\right)=\{\begin{array}{c}\frac{\left|x\right|}{x}\\ 0,\end{array}\begin{array}{c},x\ne 0\\ x=0\end{array}$

For x ≠ 0,

f(x) = -x/x = -1,   if x < 0

f(x) = x/x = 1. if x > 0

Now,

LHL = $\underset{x\to 0^{-}}{lim}f(x)=\underset{x\to 0^{-}}{lim}-x/x=-1$

RHL = $\underset{x\to 0^{+}}{lim}f(x)=\underset{x\to 0^{+}}{lim}x/x=1$

$\underset{x\to 0}{lim}f(x)=0$

Since,

LHL ≠ RHL we can say that the function is not continuous at x = 0

Only x = 0, is the point of discontinuity. 

Based on the definition, the function has discontinuity of the first kind.