$\text{ If }\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{cc}{\mathrm{x}}^2\left\{{\mathrm{e}}^{1/\mathrm{x}}\right\}& \mathrm{x}\ne 0\\ \mathrm{k}& \mathrm{x}=0\end{array}\right.$ is continuous at x=0, then which of the following is/are CORRECT?  ({.} denotes fractional part function)

$$\begin{gathered} \underset{\mathrm{x}\to 0}{lim} \mathrm{f}\left(\mathrm{x}\right)=0\left\{\because \underset{\mathrm{x}\to 0}{lim} {\mathrm{x}}^2=0\right.\text{ and }\left.\left\{{\mathrm{e}}^{1/\mathrm{x}}\right\}\text{ is a bounded function }\right\} \\ \begin{array}{l}\underset{\mathrm{x}\to 0}{lim} \frac{\mathrm{f}(0+\mathrm{x})-\mathrm{f}\left(0\right)}{\mathrm{x}}=\underset{\mathrm{x}\to 0}{lim} \mathrm{x}\left\{{\mathrm{e}}^{1/\mathrm{x}}\right\}=0\\ \therefore {\mathrm{f}}^{\mathrm{\prime }}\left(0\right)=0 \\ \mathrm{f}\left(\mathrm{x}\right)\text{ is not continuous at }\mathrm{x}={\log }_2\mathrm{e},{\log }_3\mathrm{e},\dots \text{ etc. }\end{array} \end{gathered}$$