Define F(x) as the product of two realfunctions f1(x)=x,x∈R, and f2(x)=sin⁡1/x; if x≠00; if x=0as follows F(x)=f1(x)f2(x);x≠00;x=0Statement-1: F(x) is continuous on RStatement-2: f1(x) and f2(x) are continuous on R.

# Define as the product of two realfunctions ${f}_{1}\left(x\right)=x,x\in \mathbf{R}$, and as follows $\mathrm{F}\left(x\right)=\left\{\begin{array}{ccc}{f}_{1}\left(x\right){f}_{2}\left(x\right)& ;& x\ne 0\\ 0& ;& x=0\end{array}\right\$Statement- is continuous on RStatement-) and ${f}_{2}\left(x\right)$ are continuous on.

1. A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1

2. B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for
STATEMENT-1

3. C

STATEMENT-1 is True, STATEMENT-2 is False

4. D

STATEMENT-1 is False, STATEMENT-2 is True

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### Solution:

Since $|x\mathrm{sin}1/x|\le |x|$ so $\underset{x\to 0}{lim} \mathrm{F}\left(x\right)=0=$

$F\left(0\right)$ thus $F$ is a continuous function. Since $\underset{x\to 0}{lim} \mathrm{sin}1/x$

doesn’t exist so ${f}_{2}$ is not continuous at

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