Question

Discuss the continuity and differentiablity of the function $f\left(x\right)=\sin x+\sin \left|x\right|$, $x\in R$. Draw a rough sketch of the graph of $f\left(x\right)$

• A. $f\left(x\right)$ is continuous but not differentiable at $x=0$
• B. $f\left(x\right)$ is continuous and differentiable $\forall x\in R$,
• C. $f\left(x\right)$ is not continuous and not derivable at $x=0$
• D. none of these

Answer 1

The correct option is A $f\left(x\right)$ is continuous but not differentiable at $x=0$

$f\left(x\right)=sinx+sin|x|$
$f\left(0^{+}\right)=\underset{h\to 0}{lim}\left(2sinh\right)=0$
$f\left(0^{-}\right)=\underset{h\to 0}{lim}(sin-h+sinh)=0$
Hence the function is continuous at $x=0.$
$f^{\prime }\left(0^{+}\right)=\underset{h\to 0}{lim}\frac{sinh+sinh}{h}=2$
$f^{\prime }\left(0^{-}\right)=\underset{h\to 0}{lim}\frac{sin-h+sinh}{-h}=0$
Hence the function is not differentiable at $x=0.$