Question

Discuss the continuity and differentiabilty of $f\left(x\right)=\left|\log \left|x\right|\right|$.

Answer 1

$f\left(x\right)=|\log |x||⟹|x|\ne 0$

i.e at $x=0,f\left(x\right)$ is discontinuous

Let $t=\log |x|$

$⟹f\left(x\right)=|t|$ which is not differentiable at $t=0⟹\log |x|=0⟹|x|=1⟹x=\pm 1$

$f\left(x\right)$ is not differentiable at $x=0,1,-1$