\begin{equation}
h(x)=
\begin{cases}
x^2 \sin(\frac{1}{x})&\text{ if } x\neq 0\\
0&\text{ if } x=0
\end{cases}
\end{equation}
Is the derivative of $h(x)$ continuous at $x = 0$?
How about the derivative of $k(x) = xh(x)$?
When I tried to differentiate it directly, the $\cos(1/x)$ in the result suggests that the limit of $h'(x)$ when $x\rightarrow 0$ does not exist.
However, when I use the definition to calculate the derivative, it shows that the derivative is $0$ when $x\rightarrow0$.
Which one is correct?
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