Question

Let a function be defined as $f\left(x\right)=\frac{x-\left|x\right|}{x}$. Then $f\left(x\right)$ is

• A. continuous nowhere
• B. continuous everywhere
• C. continuous for all $x$ except $x=1$
• D. continuous for all $x$ except $x=0$

Answer 1

The correct option is D continuous for all $x$ except $x=0$

$\left|x\right|=\{\begin{array}{c}x;x\ge 0\\ -x;x<0\end{array}\Rightarrow f\left(x\right)=\frac{x-\left|x\right|}{x}=\{\begin{array}{c}0;x>0\\ 2;x<0\end{array}$
The function is not defined at $x=0$. It is discontinuous at this point, but is continuous everywhere else.