Question

$f\left(x\right)=\left(\right[x]+[-x\left]\right),x\ne 3$ is continuous at $x=3,$ then the value of $f\left(3\right)$ is $($where $[.]$ denotes the greatest integer function$)$

• A. $0$
• B. $1$
• C. $-1$
• D. $2$

Answer 1

The correct option is C $-1$

Given : $f\left(x\right)=\left(\right[x]+[-x\left]\right),x\ne 3$
At $x=3,$
$L.H.L.=$
$\underset{h\to 0}{lim}f(3-h)=\underset{h\to 0}{lim}\left(\right[3-h]+[-3+h\left]\right)$
$=2+(-3)=-1$

Since $f\left(x\right)$ is continuous at $x=3$
$L.H.L.=f\left(3\right)=-1$
$\therefore f\left(3\right)=-1$