Question

Let a function $f:R\to R$ be defined as $f\left(x\right)=\{\begin{array}{cc}\sin x-e^x& \text{if}x\le 0\\ a+[-x]& \text{if}0<x<1\\ 2x-b& \text{if}x\ge 1\end{array}$

where $\left[x\right]$ is the greatest integer less than or equal to $x$. If $f$ is continuous on $R$, then $(a+b)$ is equal to

• A. $5$
• B. $3$
• C. $4$
• D. $2$

Answer 1

The correct option is B $3$

$f\left(x\right)=\{\begin{array}{cc}\sin x-e^x& \text{if}x\le 0\\ a-1-\left[x\right]& \text{if}0<x<1\\ 2x-b& \text{if}x\ge 1\end{array}$

Given: $f\left(x\right)$ is continuous everywhere
$\therefore f\left(x\right)$ is continuous at $x=0$
$\therefore \underset{x\to 0^{-}}{lim}f\left(x\right)=f\left(0\right)=\underset{x\to 0^{+}}{lim}f\left(x\right)$
$\Rightarrow –1=a-1\Rightarrow a=0$
$f\left(x\right)$ is also continuous at $x=1$
$\therefore \underset{x\to 1^{-}}{lim}f\left(x\right)=f\left(1\right)=\underset{x\to 1^{+}}{lim}f\left(x\right)$
$\Rightarrow a-1=2-b\Rightarrow -1=2-b$
$\Rightarrow b=3$
$\therefore a+b=3$