Question

Let $f\left(x\right)=\{\begin{array}{c}-(\frac{1}{\left|x\right|}-\frac{1}{x})x\ne 0\\ 0x=0\end{array}$. Discuss the continuity and differentiability at $x=0$.

Answer 1

Because $|x|=xforx\le 0|x|=-xforx<0f\left(x\right)=1ifx>3andf\left(x\right)=-1ifx<0Sof\left(x\right)=1forx>0Sof\left(x\right)=0$

$f\left(x\right)=(\frac{1}{|x|}-\frac{1}{x})=(\frac{1}{0}-\frac{1}{0})=0$

does not hold

So it is not continuity and diffentiable at $x=0$