Question

Let $f\left(x\right)=\left[x\right]$ and $g\left(x\right)=\{\begin{array}{cc}0,& x\in Z\\ x^2,& x\in R-Z\end{array}$. Then which of the following is not true ([.] represents the greatest integer function)?

• A. $\underset{x\to 1}{lim}g\left(x\right)$ exists but $g\left(x\right)$ is not continuous at $x=1$.
• B. $\underset{x\to 1}{lim}f\left(x\right)$ does not exist and $f\left(x\right)$ is not continuous at $x=1$.
• C. $gof$ is a continuous function.
• D. $gof$ is a discontinuous function.

Answer 1

Answer: (A), (B), (C)

$\underset{x\to 1}{lim}g\left(x\right)$ exists but $g\left(x\right)$ is not continuous at $x=1$.
$\underset{x\to 1}{lim}f\left(x\right)$ does not exist and $f\left(x\right)$ is not continuous at $x=1$.
$gof$ is a continuous function.

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

$\underset{x\to 1^{-}}{lim}f\left(x\right)=0$

$f\left(x\right)=1atx=1$

So limit does not exist so discontinous at 1

$g\left(x\right)=0atx\in Z$

$g\left(x\right)=x^{2}atx\in R-Z$

$\underset{x\to 1^{+}}{lim}g\left(x\right)=1$

$\underset{x\to 1^{-}}{lim}g\left(x\right)=1$

$g\left(1\right)=0$

limit exists but function is discontinuous as value at 1 is not same as limit

$g\left(f\right(x\left)\right)=g\left(\right[x\left]\right)=0forallx$

It is continuous at all x being a consatant function

Hence, option 'C' is correct.