calculus - Paradox: Is the derivative of this function continuous at $x=0$?

$$\begin{equation} h(x)= \begin{cases} x^2 \sin(\frac{1}{x})&\text{ if } x\neq 0\\ 0&\text{ if } x=0 \end{cases} \end{equation}$$
Is the derivative of $h(x)$ continuous at $x = 0$?

How about the derivative of $k(x) = xh(x)$?

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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?