Question

If the function $f$ is defined by $f(x)=\frac{x}{1+|x|}$ then at what points is $f$ differentiable

• A. everywhere
• B. At $x=\pm 1$
• C. Except at $x=0$
• D. Except at $x=0$ or $\pm 1$

Answer 1

Answer: (A)

if $x>0$, then $f(x)=\frac{x}{1+x}$ is differentiable
$f^{\prime }\left(0^{-}\right)=\underset{x\to 0}{\lim }\frac{\frac{x}{1+x}-0}{x-0}$
$=\underset{x\to 0}{\lim }\frac{1}{1-x}=1$
if $x<0$, then $f(x)=\frac{x}{1-x}$ is differentiable at $x=0$,
$f^{\prime }\left(0^{+}\right)=\underset{x\to 0}{\lim }\frac{\frac{x}{1+x}-0}{x-0}$
$=\underset{x\to 0}{\lim }\frac{1}{1+x}=1$
$f(x)$ is differentiable at $x=0$ and hence every where