Question

Let a function $f$ be defined by $f\left(x\right)=\frac{x-\left|x\right|}{x}$ for $x\ne 0$ and $f\left(0\right)=2$ then $f$ is

• A. continuous no where
• B. continuous every where
• 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 $x=0$

$f\left(x\right)=\frac{x-\mid x\mid }{x}$

$f\left(x\right)=\left\{\begin{array}{c}\frac{x-x}{x}x<0\\ \frac{x+x}{x}x\ge 0\end{array}\right\}$

At $x=0,$

L.H.L(Left Hand Side)

${lim}_{x\to 0^{-}}f\left(x\right)=0$

R.H.L(Right Hand Side)

${lim}_{x\to 0^{+}}f\left(x\right)=2$

L.H.L=R.H.L

$\therefore {lim}_{x\to 0^{-}}\ne {lim}_{x\to 0^{+}}$

$\therefore$ It is discontinous at $x=0$

$f\left(x\right)$ is discontinuous for all X except at $X=0$.