Question

The range of $a$ for which $f\left(x\right)=\text{sgn}\left(\right(\sin x)+a)$ is continuous for all $x\in R$.

• A. $[-1,1]$
• B. $(-\infty ,-1)\cup (1,\infty )$
• C. $(-1,1)$
• D. $(-\infty ,-1]\cup [1,\infty )$

Answer 1

The correct option is B $(-\infty ,-1)\cup (1,\infty )$

For $f\left(x\right)=\text{sgn}\left(\right(\sin x)+a)$ to be continuous
$\left(\right(\sin x)+a)$ must be either always positive or always negative.
For being always positive : $-1+a>0$ $\Rightarrow a>1$
For being always negative : $1+a<0$ $\Rightarrow a<-1$
$\Rightarrow a\in (-\infty ,-1)\cup (1,\infty )$