Question

If $f\left(x\right)$ is discontinuous at only $x=1$ such that $f^2\left(x\right)=4\forall xϵR$, then number of points $f\left(x\right)$ is discontinuous are

• A. $4$
• B. $6$
• C. $8$
• D. $10$

Answer 1

Answer: (B)

$6$

$f^2\left(x\right)=4\Rightarrow f\left(x\right)=\pm 2\forall xϵR$
The function is continuous everywhere for $x\ne 1$. It can have the following definitions:
$f\left(x\right)=\{\begin{array}{c}2;x\ne 1\\ -2;x=1\end{array}f\left(x\right)=\{\begin{array}{c}2;x<1\\ -2;x\ge 1\end{array}f\left(x\right)=\{\begin{array}{c}2;x\le 1\\ -2;x>1\end{array}f\left(x\right)=\{\begin{array}{c}-2;x\ne 1\\ 2;x=1\end{array}f\left(x\right)=\{\begin{array}{c}-2;x<1\\ 2;x\ge 1\end{array}f\left(x\right)=\{\begin{array}{c}-2;x\le 1\\ 2;x>1\end{array}$
So there are six such functions.