Mathematics - Continuity - Differentiability Question with Solution | TestHub

$$\begin{array}{rl}& f\circ g=f(g(x))=f(1-|x|)\\ & =-1+|1-|x|-2|\\ & =-1+|-|x|-1|=-1+||x|+1|\end{array}$$ Let $f\circ g=y$ $$\begin{array}{rl}& \therefore y=-1+||x|+1|\\ & \Rightarrow y=\{\begin{array}{ll}-1+x+1,& x\ge 0\\ -1-x+1,& x<0\end{array}\\ & \Rightarrow y=\{\begin{array}{cc}x,& x\ge 0\\ -x,& x<0\end{array}\end{array}$$ LHL at $(x=0)={\lim }_{x\to 0}(-x)=0$ RHL at $(x=0)={\lim }_{x\to 0}(x)=0$ When $x=0$, then $y=0$ Hence, LHL at $(x=0)=$ RHL at $(x=0)=$ value of $y$ at $(x=0)$ Hence $y$ is continuous at $x=0$. Clearly at all other point $y$ continuous. Therefore, the set of all points where fog is discontinuous is an empty set.