Question

Let $f\left(x\right)=\left[x\right]+[-x]$, where [x] denotes the greatest integer less than or equal to x.

Then, for any integer $m$

• A. $\underset{x\to m}{lim}f\left(x\right)=f\left(m\right)$
• B. $\underset{x\to m}{lim}f\left(x\right)\ne f\left(m\right)$
• C. $\underset{x\to m}{lim}f\left(x\right)$ does not exist
• D. None of the above

Answer 1

The correct option is B $\underset{x\to m}{lim}f\left(x\right)\ne f\left(m\right)$

$f\left(x\right)=\left[x\right]+[-x]$

$f\left(m\right)=\left[m\right]\_[-m]$

$=m-m=0$

$\underset{x\to m^{+}}{lim}f\left(x\right)=m+(-m-1)([-x]=[-\left(m^{+}\right)])$

$=-1$

$\underset{x\to m^{-}}{lim}f\left(x\right)=(m-1)+(-m)([-x]=[-\left(m^{-}\right)])$

$=-1$

LHL=RHL

$\underset{x\to m}{lim}f\left(x\right)=-1$

But $\underset{x\to m}{lim}f\left(x\right)\ne f\left(m\right)$