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 $F (x)$ as the product of two real
functions $f_{1} (x) = x, x \in R$ , and $f_{2} (x) = \{\begin{array}{cll} \sin 1 / x & ; & if x \neq 0 \\ 0 & ; & if x = 0 \end{array}$
as follows $F (x) = \{\begin{array}{ccc} f_{1} (x) f_{2} (x) & ; & x \neq 0 \\ 0 & ; & x = 0 \end{array}$
Statement- $1 : F (x)$ is continuous on R
Statement- $2 : f 1 (x$ ) and $f_{2} (x)$ are continuous on $R$ .

A
STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B
STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for
STATEMENT-1
C
STATEMENT-1 is True, STATEMENT-2 is False
D
STATEMENT-1 is False, STATEMENT-2 is True

Solution:

Since $| x \sin 1 / x | \leq | x |$ so $lim_{x \to 0} F (x) = 0 =$

$F (0)$ thus $F$ is a continuous function. Since $lim_{x \to 0} \sin 1 / x$

doesn’t exist so $f_{2}$ is not continuous at $x = 0 .$

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