Question

Let $\left[x\right]$ denotes the greatest integer less than or equal to x. If $f\left(x\right)=\left[x\sin \pi x\right]$, then $f\left(x\right)$ is?

• A. Continuous at $x=0$
• B. Continuous in $(-1,0)$
• C. Differentiable at $x=1$
• D. Differentiable in $(-1,1)$

Answer 1

The correct option is A Continuous at $x=0$

$\left[x\right]$ denotes greatest integral function.

$f\left(x\right)=\left[x\sin \pi x\right]$

$L.H.S$

$⟹{lim}_{x⟶0^{-}}\left[x\sin \pi x\right]$

$⟹{lim}_{n⟶0}\left[\right(0-n\left)\sin \pi \right(0-n\left)\right]$

$⟹{lim}_{n⟶0}\left[n\sin \pi n\right]=0$

$R.H.S$

$⟹{lim}_{x⟶0^{+}}\left[x\sin \pi x\right]$

$⟹{lim}_{n⟶0}\left[\right(0+n\left)\sin \pi \right(0+n\left)\right]$

$⟹{lim}_{n⟶0}\left[n\sin \pi n\right]=0$

$f\left(0\right)=\left[x\sin \pi x\right]=0$

Hence $L.H.S=R.H.S=f\left(0\right).$

$\therefore$ $f\left(x\right)$ is continous $x=0.$