Let a function $$f : R \to R$$ be defined as,
$$$f(x) = \begin{cases} \sin x - e^x & \text{if } x \le 0 \\ a + [-x] & \text{if } 0 < x < 1 \\ 2x - b & \text{if } x \ge 1 \end{cases}$$$
Where $$[x]$$ is the greatest integer less than or equal to $$x$$. If $$f$$ is continuous on $$R$$, then $$(a + b)$$ is equal to:

For continuity we need to check the two potential discontinuities at $$x = 0$$ and $$x = 1$$.

At $$x = 0$$: The left-hand limit is $$\lim_{x \to 0^-}(\sin x - e^x) = \sin 0 - e^0 = -1$$. For the right-hand limit, when $$0 < x < 1$$, we have $$-1 < -x < 0$$, so $$[-x] = -1$$. Thus the right-hand limit is $$\lim_{x \to 0^+}(a + [-x]) = a - 1$$. Setting equal: $$a - 1 = -1$$, giving $$a = 0$$.

At $$x = 1$$: For the left-hand limit, as $$x \to 1^-$$ with $$0 < x < 1$$, we have $$-x \to -1^+$$, so $$[-x] = -1$$. Thus the left-hand limit is $$0 + (-1) = -1$$. The right-hand value is $$2(1) - b = 2 - b$$. Setting equal: $$2 - b = -1$$, giving $$b = 3$$.

Therefore $$a + b = 0 + 3 = 3$$.