Question

Consider the function f(x) = $\left|x\right|$ in the interval -1 < x < 1. At the point x = 0, f(x) is

• A. continuous and differentiable.
• B. non-continuous and differentiable.
• C. continuous and non-differentiable.
• D. neither continuous nor differentiable.

Answer 1

The correct option is C continuous and non-differentiable.

f(x) = $\left|x\right|$

$\Rightarrow f\left(x\right)=\{\begin{array}{cc}x& x\ge 0\\ -x& x<0\end{array}$

For continuous

at $x\to 0^{+}$

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

at $x\to 0^{-}$

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

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

$\therefore$ The function is continuous

for differentiability
at $x\to 0^{-}$

$\frac{df}{dx}{\mid }_{x=0^{-}}=\frac{d}{dx}(-x)=-1$

at $x\to 0^{+}$

$\frac{df}{dx}{\mid }_{x=0^{+}}=\frac{d}{dx}\left(x\right)=1$

$\therefore$ The given function is not differentiable.

Alternative Solution: