Question

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

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

Answer 1

Answer: (B), (D)

The correct options are
B has a missing point discontinuity
D irremovable discontinuity

$f\left(x\right)=sgn(\cos 2x-2\sin x+3)$
$=sgn(1-2{\sin }^{2}x-2\sin x+3)$
$=sgn(1-2{\sin }^{2}x-2\sin x+4)$
f(x) is discontinuous when $-2{\sin }^{2}x-2\sin x+4=0$ or ${\sin }^{2}x+\sin x-2=0$
or $(\sin x-1)(\sin x+2)=0$ or $\sin x=1$
Hence f(x) is discontinuous.