Question

Let $f\left(x\right)=\{\begin{array}{cc}\left[x\right]+[-x],& x\ne 2\\ \lambda ,& x=2\end{array};$ where $[.]$ denotes the greatest integer function. If $f$ is continuous at $x=2,$ then the value of $\lambda$ is

• A. $-1$
• B. $1$
• C. $0$
• D. $2$

Answer 1

The correct option is A $-1$

$f\left(x\right)=\{\begin{array}{cc}\left[x\right]+[-x],& x\ne 2\\ \lambda ,& x=2\end{array}$
If $f$ is continuous at $x=2,$ then
$L.H.L.=R.H.L.=f\left(2\right)$

$L.H.L.=\underset{x\to 2^{-}}{lim}(\left[x\right]+[-x])$
$=\underset{h\to 0}{lim}[2-h]+[-(2-h)]$
$=\underset{h\to 0}{lim}[2-h]+[-2+h]$
$=\left(1\right)+(-2)=-1$

$R.H.L.=\underset{x\to 2^{+}}{lim}(\left[x\right]+[-x])$
$=\underset{h\to 0}{lim}[2+h]+[-(2+h)]$
$=\left(2\right)+(-3)=-1$

$f\left(2\right)=\lambda =L.H.L.=R.H.L.$
$\therefore \lambda =-1$