Question

Which one of the following is correct in respect of the function $f\left(x\right)=\frac{x^2}{\left|x\right|}$ for $x\ne 0$ and $f\left(0\right)=0$?

• A. $f\left(x\right)$ is discontinuous everywhere
• B. $f\left(x\right)$ is continuous everywhere
• C. $f\left(x\right)$ is continues at $x=0$ only
• D. $f\left(x\right)$ is discontinuous at $x=0$ only

Answer 1

Answer: (B)

$f\left(x\right)$ is continuous everywhere

$f\left(x\right)=\frac{x^2}{\left|x\right|}$ for $x\ne 0$

$\Rightarrow f\left(x\right)=\frac{x^2}{-x}=-x$ for $x<0$ and

$f\left(x\right)=x$ for $x>0$

Also, $f\left(x\right)=0$ for $x=0$

Thus, $f\left(x\right)=\left|x\right|$

Hence, $f\left(x\right)$ is continuous everywhere.

Option B is correct.