Let a function f: R → R be defined as, f(x) = \ cases x - e ^ x &...

We have,

$f$\left(x\right) = \left\{$\begin{cases}\sin x - {e}^{x} & ifx \le 0 \\ a + \left[- x\right] & if0 < x < 1 \\ 2x - b & ifx \ge 1\end{cases}$\right)

If $f\left(x\right)$ is continuous at $x = a$ then $\lim_{x \to {a}^{-}}f\left(x\right) = f\left(a\right) = \lim_{x \to {a}^{+}}f\left(x\right)$.

Continuous at $x = 0$

$f\left({0}^{+}\right) = f\left({0}^{-}\right)$

$⇒ a - 1 = 0 - {e}^{0}$

$⇒ a = 0$

Continuous at $x = 1$

$f\left({1}^{+}\right) = f\left({1}^{-}\right)$

$⇒ 2\left(1\right) - b = a + \left(- 1\right)$

$⇒ b = 2 - a + 1$

$⇒ b = 3$

$∴ a + b = 3$