Question

If $f\left(x\right)=\{\begin{array}{c}\left[x\right]+[-x],x\ne 2\\ \lambda ,x=2\end{array}$, $f$ is continuous at $x=2$ then $\lambda$ is (where $[\cdot ]$ denotes greatest integer)-

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

Answer: (A)

$-1$

$\underset{x⟶2^{+}}{lim}f\left(x\right)=\left[2^{+}\right]+[-2^{+}]=2-3=-1$
$\underset{x⟶2^{-}}{lim}f\left(x\right)=\left[2^{-}\right]+[-2^{-}]=1-2=-1$
For $f\left(x\right)$ to be continuous
$\underset{x⟶2^{+}}{lim}f\left(x\right)=\underset{x⟶2^{-}}{lim}f\left(x\right)=f\left(2\right)=\lambda$
$\Rightarrow \lambda =-1$