If $\text{⨍ (x)=}\left\{\begin{array}{ll}{\mathrm{x}}^2\log (\cos \mathrm{x})& , \mathrm{x}\ne 0\\ 0& , \mathrm{x}=0\end{array}\right.$ then $\text{⨍ }$ is

$$\begin{gathered} \text{⨍ (x)=}\left\{\begin{array}{ll}{\mathrm{x}}^2\log (\cos \mathrm{x})& , \mathrm{x}\ne 0\\ 0& , \mathrm{x}=0\end{array}\right. \\ \underset{x\to 0}{\lim }\text{f}\left(x\right)\text{=}\underset{x\to 0}{\lim }{\mathrm{x}}^2\log (\cos \mathrm{x})=0 \\ f is continuous at x=0 \\ f\text{'}\left(0\right)=\underset{x\to 0}{\lim }\frac{{\mathrm{x}}^2\log (\cos \mathrm{x})-0}{x-0}=\underset{x\to 0}{\lim }\mathrm{x} \log (\cos \mathrm{x})=0 \\ \therefore f is differentiable at x=0 \end{gathered}$$