Question

Which one of the following function is continuous everywhere in its domain but has at least one point where it is not differentiable?

• A. $f\left(x\right)=x^{1/3}$
• B. $f\left(x\right)=\frac{|x|}{x}$
• C. $f\left(x\right)=e^{-x}$
• D. $f\left(x\right)=\tan x$

Answer 1

The correct option is A $f\left(x\right)=x^{1/3}$

To find: To check for continuity and differentiability of each

(A): $f\left(x\right)=x^{\frac{1}{3}}\to$ continues every where in its own domain

$f\left(x\right)=\frac{1}{{3x}^{\frac{2}{3}}}\to$ differentiabity not possible at n=0

(B) $f\left(x\right)=\frac{\left|x\right|}{x}\to$ discontinous at x=o

Since for x<0$\to$ f(x)= -1 and

$x\ge 0\to$ f(x)= +1

(C): $f\left(x\right)=e^{-x}\to$ continous everywhere in its domain

$f\left(x\right)={-e}^{-x}\to$ differentiable everywhere

(D): $f\left(x\right)=\tan x\to$ discontinous at

$x=(2k-1)\frac{\pi }{2}$ where $k\in I$