If $\text{⨍ (x)=}\left\{\begin{array}{cc}{\mathrm{x}}^{\mathrm{p}}\cos \frac{1}{\mathrm{x}},& \mathrm{x}\ne 0\\ 0 ,& \mathrm{x}=0\end{array}\right.\begin{array}{} \\ \end{array}$ then at $\text{x=0, ⨍ (x)}$ is

$$\begin{gathered} \mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}{\mathrm{x}}^{\mathrm{p}} \cos \frac{1}{\mathrm{x}} ; \mathrm{x}\ne 0\\ 0 ; \mathrm{x}=0\end{array}\right. \\ \underset{\mathrm{x}\to 0}{\lim }\mathrm{f}\left(\mathrm{x}\right)=\underset{\mathrm{x}\to 0}{\lim }{\mathrm{x}}^{\mathrm{p}}\left(\cos \frac{1}{\mathrm{x}}\right)=0 \mathrm{if} \mathrm{p}>0 \\ \therefore \mathrm{f} \mathrm{is} \mathrm{continuous} \mathrm{if} \mathrm{x}>0 \end{gathered}$$

$$\begin{gathered} {\mathrm{f}}^1\left(0\right)=\underset{\mathrm{x}\to 0}{\lim }\frac{\mathrm{f}\left(\mathrm{x}\right)-\mathrm{f}\left(0\right)}{\mathrm{x}-0}=\underset{\mathrm{x}\to 0}{\lim }\frac{{\mathrm{x}}^{\mathrm{p}} \cos {\displaystyle \frac{1}{\mathrm{x}}}-0}{\mathrm{x}-0} \\ =\underset{\mathrm{x}\to 0}{\lim }{\mathrm{x}}^{\mathrm{p}-1}\left(\cos \frac{1}{\mathrm{x}}\right)=0 \\ \mathrm{if} \mathrm{p}-1>0 \\ \therefore \mathrm{f} \mathrm{is} \mathrm{differentiable} \mathrm{if} \mathrm{p}>1 \\ \Rightarrow \mathrm{f} \mathrm{is} \mathrm{continuous} \mathrm{if} \mathrm{p}>0 \mathrm{and} \mathrm{differentiable} \mathrm{if} \mathrm{p}>1 \end{gathered}$$