Let $$f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right.$$ Then at $x=0$

Question

Let $$f(x) = \left\{ {\matrix{ {{x^2}\sin \left( {{1 \over x}} \right)} & {,\,x \ne 0} \cr 0 & {,\,x = 0} \cr } } \right.$$

Then at $x=0$

$f$ is continuous but $f'$ is not continuous

$f$ and $f'$ both are continuous

$f$ is continuous but not differentiable

$f'$ is continuous but not differentiable

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