Mathematics - Continuity - Differentiability Question with Solution | TestHub

Let$f,g:\mathit{R}\to \mathit{R}$be functions defined by
$f\left(x\right)=\left\{\begin{array}{ll}\left[x\right]& , x<0\\ \left|1-x\right|& , x\ge 0\end{array}\right.$and
$g\left(x\right)=\left\{\begin{array}{ll}{e}^x-x,& x<0\\ {\left(x-1\right)}^2-1,& x\ge 0\end{array}\right.$

$\Rightarrow fog\left(x\right)=\left\{\begin{array}{l}\left[g\left(x\right)\right], g\left(x\right)<0\\ \left|1-g\left(x\right)\right|, g\left(x\right)\ge 0\end{array}\right.$

After taking intersection we get,

$\Rightarrow fog\left(x\right)=\left\{\begin{array}{l}\left|1+x-e^x\right|, x<0\\ \left[{\left(x-1\right)}^2\right]-1, 0\le x<2\\ \left|2-{\left(x-1\right)}^2\right|, x\ge 2\end{array}\right.$ 

Now, checking continuity at $x=0$ and at $x=2$

We get, $fog\left(0^{+}\right)=-1$, $fog\left(0\right)=0$

Hence, discontinuous at $x=0$

And, $fog\left(2^{+}\right)=1$, $fog\left(2^{-}\right)=-1$

Hence, discontinuous at $x=2$