Question

If $f\left(x\right)=\{\begin{array}{c}x\sin \frac{1}{x}\text{else where}\\ 0x=0\end{array},$
then $f\left(x\right)$ is

• A. not continuous at $x=0$
• B. continuous but non-differentiable at $x=0$
• C. differentiable at $x=0$ and $f^{\prime }\left(0\right)=1$
• D. differentiable at $x=0$ and $f^{\prime }\left(0\right)=0$

Answer 1

The correct option is B continuous but non-differentiable at $x=0$

$\underset{x\to 0}{lim}f\left(x\right)=\underset{x\to 0}{lim}\left(x\sin \frac{1}{x}\right)=0$
Therefore, $f\left(x\right)$ is continuous.
Checking Differentiability
$\underset{x\to 0}{lim}\frac{f\left(x\right)-f\left(0\right)}{x-0}=\underset{x\to 0}{lim}\frac{\left(x\sin \frac{1}{x}\right)-0}{x-0}$
$=\underset{x\to 0}{lim}\left(\sin \frac{1}{x}\right)=$ indeterminant
So, $f\left(x\right)$ is non differentiable..