Question

If $f\left(x\right)=\{\begin{array}{c}x+\left\{x\right\}+x\sin \left\{x\right\};x\ne 0\\ 0;x=0\end{array}$ where $\left\{x\right\}$ denotes the fractional part function, then

• A. $f$ is continuous and differentiable at $x=0$
• B. $f$ is continuous but not differentiable at $x=0$
• C. $f$ is continuous and differentiable at $x=2$
• D. None of these

Answer 1

Answer: (D)

None of these

$f^{\prime }\left(0^{+}\right)=\underset{h\to 0}{lim}\frac{h+h-\left[h\right]-h\sin (h-\left[h\right])}{h}=\underset{h\to 0}{lim}\frac{2h+h\sin h}{h}=2$
$f^{\prime }\left(0^{-}\right)=\underset{h\to 0}{lim}\frac{-h-h-[-h]-h\sin (-h-[-h])}{-h}$
$=\underset{h\to 0}{lim}\frac{-2h-h\sin (-h+1)}{-h}=\underset{h\to 0}{lim}-2+\frac{1}{h}-\sin (1-h)$
$\Rightarrow$ LHD does not exist
Hence, the function is not differentiable and discontinuous at $x=0$. Similarly for $x=2$