Question

If $f\left(x\right)=sgn(\cos 2x-2\sin x+3)$, where $sgn\left(\right)$ is the signum function, then $f\left(x\right)$-

• A. is continuous over its domain
• B. has a missing point discontinuity
• C. has isolated point discontinuity
• D. has irremovable discontinuity

Answer 1

Answer: (C)

has isolated point discontinuity

$f\left(x\right)=sgn(\cos 2x-2\sin x+3)$
$=sgn(1-2{\sin }^{2}x-2\sin x+3)$

$=sgn(4-2\sin x-2{\sin }^{2}x)$

$=sgn\left[2\right(2-\sin x\left)\right(1-\sin x\left)\right]$

Let $g\left(x\right)=2(2-\sin x)(1-\sin x)$
Clearly $g\left(x\right)\ge 0,$ since $-1\le \sin x\le 1$

$g\left(x\right)=0\Rightarrow (1-\sin x)=0\Rightarrow x=(2n+1)\frac{\pi }{2}$, $n\in I$
Thus $f\left(x\right)=\{\begin{array}{c}0\text{when}x=(2n+1)\frac{\pi }{2}\\ 1\text{otherewise}\end{array}$

Hence f is discontinuous at $x=(2n+1)\frac{\pi }{2}$
which is clearly isolated type of discontinuity.
Hence,the correct answer is option 'C'.