Let f,g:R→R be functions defined by
f(x)={\begin{matrix}[x] & ,x<0 \\ |1-x| & ,x\geq 0\end{matrix} and
g(x)={\begin{matrix}{e}^{x}-x, & x<0 \\ {(x-1)}^{2}-1, & x\geq 0\end{matrix}\Rightarrow fog(x)={\begin{matrix}[g(x)],g(x)<0 \\ |1-g(x)|,g(x)\geq 0\end{matrix}
After taking intersection we get,
\Rightarrow fog(x)={\begin{matrix}|1+x-{e}^{x}|,x<0 \\ [{(x-1)}^{2}]-1,0\leq x<2 \\ |2-{(x-1)}^{2}|,x\geq 2\end{matrix}
Now, checking continuity at x=0 and at x=2
We get, fog(0+)=−1, fog(0)=0
Hence, discontinuous at x=0
And, fog(2+)=1, fog(2−)=−1
Hence, discontinuous at x=2