Question

Consider the following in respect of the function $f\left(x\right)=|x-3|$ :
1. $f\left(x\right)$ is continuous at $x=3$
2. $f\left(x\right)$ 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

The correct option is C Both 1 and 2

$\underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{x\to 3^{-}}{lim}-(x-3)=-(3-3)=0$

$\underset{x\to 3^{+}}{lim}f\left(x\right)=\underset{x\to 3^{+}}{lim}(x-3)=3-3=0$

$f\left(3\right)=|3-3|=0$

LHL$=$RHL $=f\left(x\right)\Rightarrow$continuous at $x=3$

$f^{\prime }\left(x\right)=\{\begin{array}{c}-1;x<3\\ 0;x=3\\ 1;x>3\end{array}$

$f^{\prime }\left(x\right)=-1$ for $x<3$

$\Rightarrow f^{\prime }\left(x\right)=-1$ for $x=0$

$\Rightarrow f\left(x\right)$ is differentiable at $x=0$