Question

Let $g\left(x\right)=\left[x\right]$, where $[.]$ represents greatest integer function. Then the function $f\left(x\right)=\left(g\right(x){)}^2-g(x)$ is discontinuous at :

• A. $x\in R$
• B. $x\in Z-\left\{1\right\}$
• C. $x\in Z$
• D. $x\in Z-\{0,1,-1\}$

Answer 1

The correct option is B $x\in Z-\left\{1\right\}$

Given $g\left(x\right)=\left[x\right]$
$\Rightarrow$$f\left(x\right)=\left(g\right(x){)}^2-g(x)=[x{]}^2-\left[x\right]$
Now, greatest integer function is discontinuous at every integer so $f\left(x\right)$ can be discontinuous at every integer.
But, $\underset{x\to 1^{-}}{lim}f\left(x\right)=[1^{-}{]}^2-[1^{-}]=0-0=0$
$\underset{x\to 1^{+}}{lim}f\left(x\right)=[1^{+}{]}^2-[1^{+}]=1-1=0$
and $f\left(1\right)=0$
Thus $f\left(x\right)$ is discontinuous at every integer except $x=1$.