Question

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

Answer 1

$f\left(x\right)=\sin \left(\right\{2^x+\left[2^x\right]+\left[3^{–x}\right]\left\}\right)$
$\Rightarrow f\left(x\right)=\sin \left\{2^x\right\}(\because \left[2^x\right],\left[3^{-x}\right]\in Z)$

$\therefore f\left(x\right)$ is discontinuous for all $x$ where $2^x$ is an integer.
For $x\in [0,4],$ $2^x\in [1,16]$
At end points, we will be checking one sided continuity.
At $x=0,$ $f\left(x\right)$ is right continuous.
At $x=4,$ $f\left(x\right)$ is not left continuous.
Hence, there are $15$ points of discontinuity.