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

Question:

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 $\mathbb{R}$.

  1. Statemen-1 is false, statement-2 is true.

  2. Statemen-1 is true,statement-2 is true;Statement-2 is correct explanation for statement1.

  3. Statement- 1 is true, statement- 2 is true, statement- 2 is not a correct explanation for statement 1

  4. Statement-1 is true, statement-2 is false


Correct Option: , 4

Solution:

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