Question

Statement $1:$ $f\left(x\right)=\sin x+\left[x\right]$ is discontinuous at $x=0,$ where $[.]$ denotes the greatest integer function.
Statement $2:$ If $g\left(x\right)$ is continuous and $h\left(x\right)$ is discontinuous at $x=a,$ then $g\left(x\right)+h\left(x\right)$ will necessarily be discontinuous at $x=a.$

• A. Both the statements are true and statement $2$ is the correct explanation of statement$1$.
• B. Both the statements are true and statement $2$ is not the correct explanation of statement $1$.
• C. statement $1$ is true andstatement $2$ is false
• D. statement $1$ is false and statement $2$ is true

Answer 1

The correct option is A Both the statements are true and statement $2$ is the correct explanation of statement$1$.

$\underset{x\to 0^{+}}{lim}(\sin x+[x\left]\right)=0,\underset{x\to 0^{-}}{lim}(\sin x+[x\left]\right)=-1$
Thus, limit does not exist, hence $f\left(x\right)$ is discontinuous at $x=0$

We know that addition or subtraction of a continuous and discontinuous function will be discontinuos.
Here $\sin x$ is continuous at $x=0$
but $\left[x\right]$ is discontinuous at $x=0$
$\Rightarrow \sin x+\left[x\right]$ is discontinuous at $x=0$
$\therefore$ Statement $2$ is also correct and correct explanation of statement $1.$