Question

A point $(x,y)$, where function $f\left(x\right)=\left[\sin \left[x\right]\right]$ in $(0,2\pi )$ is not continuous , is ($[.]$ denotes greatest integer$\le x$).

• A. $(3,0)$
• B. $(2,0)$
• C. $(1,0)$
• D. $(4,-1)$

Answer 1

The correct option is D $(4,-1)$

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

At $1\le x<2,\left[\sin \left[x\right]\right]=\left[\sin 1\right]$ lies between $0$ and $1$

At $2\le x<3,\left[\sin \left[x\right]\right]=\left[\sin 2\right]$ lies between $0$ and $1$

At $3\le x<4,\left[\sin \left[x\right]\right]=\left[\sin 3\right]$ lies between $0$ and $1$

At $4\le x<5,\left[\sin \left[x\right]\right]=\left[\sin 4\right]<1$

At $5\le x<6,\left[\sin \left[x\right]\right]=\left[\sin 5\right]<1$

At $6\le x<7,\left[\sin \left[x\right]\right]=\left[\sin 6\right]<1$

From the graph we observe that there is a discontinuity from $(4,0)$

$\therefore$ the function is discontinuous at $(4,-1)$