Question

Let $f\left(x\right)=\text{sgn}\left(x\right).\sin x,$ where $\text{sgn}\left(x\right)$ is signum of $x$. Which of the following is INCORRECT ?

• A. $f\left(x\right)$ is continuous everywhere.
• B. $f\left(x\right)$is an even function
• C. $f\left(x\right)$ is non-periodic.
• D. $f\left(x\right)$ is differentiable for all $x$ expect $x=0.$
• E. $f\left(x\right)$ is non-monotonic.

Answer 1

Answer: (A), (B), (D)

$f\left(x\right)$ is continuous everywhere.
$f\left(x\right)$is an even function
$f\left(x\right)$ is differentiable for all $x$ expect $x=0.$

$f\left(x\right)=\{\begin{array}{cc}-\sin \left(x\right)& x<0\\ 0& x=0\\ \sin \left(x\right)& x>0\end{array}$
Hence $f\left(x\right)$ is continuous for all $x$ and the graph of $f\left(x\right)$ is symmetric about $y$ axis.
However there is a sharpness in the graph at $x=0.$
Thus the function is not differentiable at $x=0.$