Question

Let $f\left(x\right)=\{\begin{array}{c}\sin x,x\ge 0\\ -\sin x,x<0\end{array}$.

Then $f\left(x\right)$ is?

• A. continuous at $x=0$
• B. differentiable at $x=0$
• C. discontinuous at $x=0$
• D. not differentiable at $x=0$

Answer 1

Answer: (A), (D)

The correct options are
A continuous at $x=0$
D not differentiable at $x=0$

In the given function

$L.H.L=R.H.L=f\left(x\right)$, which is a continuous function at $x=0$

$f^{\prime }\left(x\right)=\cos x$ for $x\ge 0$

$f^{\prime }\left(x\right)=-\cos x$ for $x<0$

$L.H.D=1$ and $R.H.L=-1$, not differentiable at $x=0$

Hence, options 'A' and 'D' are correct.