Question

If $f\left(x\right)=\mathrm{sgn}(2\sin x+a)$ is continuous for all $x\in R$, then the possible values of $a$ are

• A. $a\in R$
• B. $a\in (-\infty ,-2)\cup (2,\infty )$
• C. $a\in (-2,2)$
• D. $a\in (0,\infty )$

Answer 1

The correct option is B $a\in (-\infty ,-2)\cup (2,\infty )$

We know that,
$-1\le \sin x\le 1\Rightarrow a-2\le 2\sin x+a\le a+2$

We know that $\mathrm{sgn}\left(x\right)$ is continuous for $x>0$ or $x<0$

So, for $f\left(x\right)=\mathrm{sgn}(2\sin x+a)$ to be continuous,

either $a-2>0$ or $a+2<0$
$\Rightarrow a>2$ or $a<-2$

So, $a\in (-\infty ,-2)\cup (2,\infty )$