Question

Discuss the continuity of the following function, at $x=0$.
$f\left(x\right)=\frac{x}{|x|}$, for $x\ne 0$
$=1$, for $x=0$

Answer 1

Left hand limit $=\underset{x\to 0^{-}}{lim}\frac{x}{|x|}=\underset{x\to 0^{-}}{lim}\frac{x}{-x}=-1$

And right hand limit $=\underset{x\to 0^{+}}{lim}\frac{x}{|x|}=\underset{x\to 0^{+}}{lim}\frac{x}{x}=1$

Clearly LHL and RHL are not same

Hence above function is not continuous at $x=0$

Note: $|x|=\{\begin{array}{c}x,x\ge 0\\ -x,x<0\end{array}$