Question

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

• A. $(-\infty ,\infty ).$
• B. $(-\infty ,0).$
• C. $(-\infty ,1).$
• D. $(-1,\infty ).$

Answer 1

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

for $x>0$, $f\left(x\right)=\frac{x}{1+x}$
and for $x<0$, $f\left(x\right)=\frac{x}{1-x}$
Consider $x=0$ for continuity:
Since, $f(0-h)=f(0+h)=f\left(0\right)$
Therefore, f is continuous at $x-0$
and thus continuous in R
For differentiability :
$Lt\frac{f(0-h)-f\left(0\right)}{-h}=Lt\frac{f(0+h)-f\left(0\right)}{h}=1$
Therefore, it is differential at $x=0$
and thus differential at R
Ans: A