Let [ t ] denote the greatest integer ≤ t . The number of points...

As $[ . ]$ is discontinuous at integers so we will check at $x = - 1, 0, 1$ only

at $x = - 1$

$[ - {1}^{-} ] = - 2$ and $[ - {1}^{+} ] = - 1$

we have

$LHL = ( - 2 ) \cdot ( 0 ) + \sin \left(\pi\right) + 1 = 1$

$RHL = ( - 1 ) \cdot ( 0 ) + \sin \left(\dfrac{\pi}{2}\right) - 0 = 1$

$f ( - 1 ) = 1 ⇒$ Continuous at $x = - 1$

Again at $x = 0$

$[ {0}^{-} ] = - 1 \text{ and } [ {0}^{+} ] = 0$

$LHL = ( - 1 ) \cdot ( 1 ) + \sin \left(\dfrac{\pi}{2}\right) - 0 = 0$

$RHL = 0 + \sin \left(\dfrac{\pi}{3}\right) - 1 \ne LHL$

Hence, discontinuous at $x = 0$

Again at $x = 1$,

$[ {1}^{-} ] = 0 \text{ and } [ {1}^{+} ] = 1$

We have

$LHL = 0 + \sin \left(\dfrac{\pi}{3}\right) - 1$

$RHL = 0 + \sin \left(\dfrac{\pi}{4}\right) - 2 \ne LHL$

Hence, discontinuous at $x = 1$

Hence, discontinuous at exactly $2$ points