Question

The set of all points where the function $f(x)=\frac{x}{1+|x|}$ is differentiable is

• A. $(-\infty ,\infty )$
• B. $(0,\infty )$
• C. $(-\infty ,0)\cup (0,\infty )$
• D. $(-\infty ,0)$

Answer 1

The correct option is A $(-\infty ,\infty )$

$f\left(x\right)=\frac{x}{1+|x|}$

For $x\ge 0$, $f\left(x\right)=\frac{x}{1+x}$

For $x<0$, $f\left(x\right)=\frac{x}{1-x}$

$f\left(x\right)$ is continuous and differentiable at all points except perhaps at $x=0$.

Hence, we will check the differentiability at $x=0$.

As $x\to 0$, $f\left(x\right)\to 0$. $f\left(0\right)=0$.

Hence, $f\left(x\right)$ is continuous at $x=0$.

$RHD=\frac{1+x-x}{(1+x{)}^2}=1$ {see that $f\left(x\right)=\infty$ at $x=-1$ but

$-1\notin [0,\infty )$}

$LHD=\frac{1-x+x}{(1-x{)}^2}=1$

Hence, $f\left(x\right)$ is differentiable at $x=0$.

Hence, option A is correct.