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  4. Let a function f: R arrow R be defined as f(x)= begincases sin x-ex text if x ≤ 0 a+[-x] text if 0 < x < 1 2 x-b text if x ≥ 1 endcases Where [x] is the greatest integer less than or equal to x. If f is continuous on R, then (a+b) is equal to:

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

JEE MainJEE Main 2021Continuity and Differentiability

Solution:

Continuous at $x=0$
$f\left(0^{+}\right)=f\left(0^{-}\right) \Rightarrow a-1=0-e^{0} $
$\Rightarrow a=0$
Continuous at $x=1$
$f\left(1^{+}\right)=f\left(1^{-}\right)$
$\Rightarrow 2(1)-b=a+(-1)$
$\Rightarrow b=2-a+1 \Rightarrow b=3$
$\therefore a+b=3$