Question

If $f\left(x\right)=\{\begin{array}{cc}{x}^{a}sin\left(\frac{1}{x}\right):& x\ne 0\\ 0:& x=0\end{array}$ is continuous but non differentiable at x = 0 then:

• A. $aϵ(-1,0)$
  
• B. $aϵ(0,2]$
  
• C. $aϵ(0,1]$
  
• D. $aϵ[1,2)$

Answer 1

The correct option is C $aϵ(0,1]$


$\underset{x\to 0}{lim}f\left(x\right)=x^{a}sin\frac{1}{x}$ exists and equal to 0 when $a>0$.
$\therefore$ f(x) will be continuous for $aϵ(0,\infty )$.
$f^{\prime }\left(0\right)=\underset{x\to 0}{lim}\frac{h^{a}sin\left(\frac{1}{h}\right)}{h}=\underset{x\to 0}{lim}{h}^{a-1}sin\left(\frac{1}{h}\right)$ will not exist when $a-1\le 0$ i.e., $a\le 1$.
$\therefore$ f(x) is not differentiable for $aϵ(-\infty ,1]$.
Hence, f(x) will be continuous and not differentiable in (0, 1]