Let f, g: R → R be functions defined by f(x) = \ cases [x] & , x...

⇒ fog\left(x\right) = \left\{$\begin{array}{ll}\left[g\left(x\right)\right], g\left(x\right) < 0 \\ \left|1 - g\left(x\right)\right|, g\left(x\right) \ge 0\end{array}$\right)</p><p>After taking intersection we get,</p><p>⇒ fog\left(x\right) = \left\{$$\begin{array}{ll}\left|1 + x - {e}^{x}\right|, x < 0 \\ \left[{\left(x - 1\right)}^{2}\right] - 1, 0 \le x < 2 \\ \left|2 - {\left(x - 1\right)}^{2}\right|, x \ge 2\end{array}$$\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 $x = 2$