Question

A function $f:R\to R$ is defined as $f\left(x\right)=x^2$ for $x\ge 0$ and $f\left(x\right)=-x$ for $x<0$.
Consider the following statements in respect of the above function:
1. The function is continuous at $x=0$.
2. The function is differentiable at $x=0$.
Which of the above statements is/are correct?

• A. 1 only
• B. 2 only
• C. Both 1 and 2
• D. Neither 1 nor 2

Answer 1

Answer: (A)

1 only

For $x=0$,

$\underset{x\to 0^{-}}{lim}f\left(x\right)=\underset{x\to 0^{}}{lim}(-x)=0$

$\underset{x\to 0^{+}}{lim}f\left(x\right)=\underset{x\to 0}{lim}{x}^2=0$

Both the limits are same, so the given function is continuous at $x=0$.

Now, left hand derivative at $x=0$ will be $\frac{d}{dx}(-x)=-1$

and right hand derivative at $x=0$ will be $\frac{d}{dx}\left(x^2\right)=2x=2\left(0\right)=0$

They are not equal, so the function is not differentiable at $x=0$.

Hence, A is correct.