Question:
Define $f(x)$ as the product of two real functions $f_1(x) = x, x \in$ R, and $f_2(x) =$ $ \begin{cases} sin \frac{1}{x} , & \text{If x $\ne $ 0} \\ 0, & \text{If x = 0} \end{cases}$
as follows :
$f(x) = \begin{cases} f_1(x).f_2(x) , & \text{If x $\ne $ 0} \\ \quad 0, & \text{If x = 0} \end{cases}$
$f(x)$ is continuous on R.
$f_1(x)$ and $f_2(x)$ 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{cases} sin \frac{1}{x} , & \text{If x $\ne $ 0} \\ 0, & \text{If x = 0} \end{cases}$
as follows :
$f(x) = \begin{cases} f_1(x).f_2(x) , & \text{If x $\ne $ 0} \\ \quad 0, & \text{If x = 0} \end{cases}$
$f(x)$ is continuous on R.
$f_1(x)$ and $f_2(x)$ are continuous on R.
Updated On: Jul 5, 2022
- Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
- Statement-1 is true, Statement-2 is true; Statement-2 is NOT a correct explanation for Statement-1
- Statement-1 is true, Statement-2 is false
- Statement-1 is false, Statement-2 is true
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The Correct Option is B
Solution and Explanation
$F(x) = \begin{cases} x \,sin(1/ x) , & \text{ x $\ne $ 0} \\ \quad 0, & \text{ x = 0} \end{cases}$
at $x = 0$
$LHL = \displaystyle\lim_{h\to0^{+}}\left\{-h\,sin \left(-\frac{1}{h}\right)\right\}$
$= 0 ? a$ finite quantity between - 1 and 1
$RHL = \displaystyle\lim_{h\to0^{+}} h sin \frac{1}{h}$
$= 0$
$f\left(0\right) = 0$
$\therefore f\left(x\right)$ is continuous on R.
$f_{2}\left(x\right)$ is not continuous at $x = 0$
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Concepts Used:
Continuity & Differentiability
Definition of Differentiability
f(x) is said to be differentiable at the point x = a, if the derivative f ‘(a) be at every point in its domain. It is given by
Definition of Continuity
Mathematically, a function is said to be continuous at a point x = a, if
It is implicit that if the left-hand limit (L.H.L), right-hand limit (R.H.L), and the value of the function at x=a exist and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is unspecified or does not exist, then we say that the function is discontinuous.