Question

Number of points of discontinuity of the function $f\left(x\right)=\sin \left(\right\{2^x+\left[2^x\right]+\left[3^{-x}\right]\left\}\right)$ for $x\in [0,4]$ is $($where $[.]$ and $.$ denotes the greatest integer and fractional part function respectively$)$

• A. 15.0
• B. 15.00
• C. 15

Answer 1

Answer: (A), (B), (C)

$f\left(x\right)=\sin \left(\right\{2^x+\left[2^x\right]+\left[3^{-x}\right]\left\}\right)$
$\Rightarrow f\left(x\right)=\sin \left\{2^x\right\}(\because [2^x\left]\&\right[3^{-x}]$ are integers)
$\therefore f\left(x\right)$ is discontinuous $\forall x$ where $2^x$ is an integer expect at $x=0$ and for $x\in [0,4]$
$\Rightarrow 2^x\in [1,16]$
Hence, $f\left(x\right)$ is discontinuous at $15$ points.