Let $f(x)=\left\{\begin{matrix}\frac{|x|}{x},&x≠0\\1,&x=0\end{matrix}\right.$ and $g(x)=\left\{\begin{matrix}x\sin(\frac{1}{x}),&x≠0\\1,&x = 0\end{matrix}\right.$

Then at the origin, which one of the following is true?

Neither f(x) nor g(x) is continuous

The correct answer is Option (2) → g(x) is continuous, but f(x) is not continuous

Given

$f(x)=\frac{|x|}{x},\; x\ne0 \;\; \text{and} \;\; f(0)=1$

$g(x)=x\sin\left(\frac{1}{x}\right),\; x\ne0 \;\; \text{and} \;\; g(0)=1$

Check continuity of $f(x)$ at $x=0$:

$\lim_{x\to0^{+}} f(x)=\lim_{x\to0^{+}} \frac{|x|}{x}=1$

$\lim_{x\to0^{-}} f(x)=\lim_{x\to0^{-}} \frac{|x|}{x}=-1$

Left and right limits are not equal, so limit does not exist.

Thus $f(x)$ is not continuous at $0$.

Check continuity of $g(x)$ at $0$:

$\lim_{x\to0} x\sin\left(\frac{1}{x}\right)=0$

But $g(0)=1$

So $\lim_{x\to0} g(x)\ne g(0)$

Thus $g(x)$ is also not continuous at $0$.

Correct option: Neither $f(x)$ nor $g(x)$ is continuous.