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Question

$$f(x)=\left\{\begin{matrix}\dfrac{x}{|x|}, & x\neq 0\\ 0, & x=0\end{matrix}\right.$$.


Solution

$$f(x)=\left\{\begin {matrix}\cfrac{x}{|x|},&x\ne0\\0,& x=0\end{matrix}\right.$$
$$f(x)=0$$
$$f(0^-)=\lim_{n\to0}f(0-h)=-1$$
$$f(0^+)=1$$
So, $$f(x)$$ is not continuous at $$x=0$$
$$f(x)=\left\{\begin {matrix}-1,&x<0\\0,& x=0\\1,&x>0\end{matrix}\right.$$

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