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Define f(x) as the product of two real f...

Define `f(x)` as the product of two real functions `f_(1)(x)=x, x epsilon R` and `f_(2)(x)={("sin"1/x, "if", x!=0),(0 , "if", x=0):}` as follows `f(x)={(f_(1)(x).f_(2)(x), "if", x!=0),(0, "if", x=0):}` Statement 2: `f_(1)(x)` and `f_(2)(x)` are continuous on IR.

A

1

B

2

C

3

D

4

Text Solution

Verified by Experts

The correct Answer is:
C
Clearly, `F(x)={{:(,x "sin"(1)/(x),x ne 0),(,0, x=0):}`
`therefore underset(x to 0)lim F(x)=underset(x to 0)lim x sin (1)/(x)=0=F(0)`
So, F(x) is continuous at x=0
Hence, statemetn-2 is correct
Statement-2: is incorrect as `f_(2)` (x) is not continuous at x=0 because `underset(x to 0)lim f_(2)(x)` does not exist
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