Define F(x) as the product of two real functions

Define $\mathrm{F}(\mathrm{x})$ as the product of two real functions $f_{1}(x)=x, x \in \mathbb{R}$, and $f_{2}(x)=\left\{\begin{array}{ccc}\sin \frac{1}{x}, & \text { if } & x \neq 0 \\ 0, & \text { if } & x=0\end{array}\right.$ as follows : $F(x)=\left\{\begin{array}{cll}f_{1}(x) \cdot f_{2}(x) & \text { if } & x \neq 0 \\ 0, & \text { if } & x=0\end{array}\right.$

Statement-1 : $F(x)$ is continuous on $\mathbb{R}$.

Statement-2 : $\mathrm{f}_{1}(\mathrm{x})$ and $\mathrm{f}_{2}(\mathrm{x})$ are continuous on $\mathbb{R}$.