Define a function $f:\mathbb{R}\rightarrow \mathbb{R}$ as $f(x)=\begin{cases}\frac{x}{|x|}& \text{if}\hspace{.2cm} x \neq 0\\ 0,& \text{otherwise} \end{cases}$. Then $f$ is

Discontinuous at 0

The correct answer is Option (2) → Discontinuous at 0

$f:R\rightarrow R$ as $f(x)=\begin{cases}\frac{x}{|x|}& \text{if}\hspace{.2cm} x \neq 0\\ 0,& \text{otherwise} \end{cases}$

$\lim\limits_{x \to 0^-}\frac{x}{-x}=-1$

$\lim\limits_{x \to 0^+}\frac{x}{x}=1$

$∴\lim\limits_{x \to 0^-}f(x)≠\lim\limits_{x \to 0^+}f(x)$

Hence, this is discontinuous.