Question

Consider the function $f(x)=|x|$ in the interval $-1\le x\le 1$. 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

Explanation:
Given,

$f(x)=|x|$$|x|=x\text{ for }x\ge 0$$|x|=-x\text{ for }x<0$At $x=0$Left limit $=0$, Right limit $=0,f(0)=0$ASLeft limit = Right limit = Function value $=0$$\therefore |x|$ is continuous at $x=0$.NowLeft derivative $($ at $x=0)=-1$Right derivative $($ at $x=0)=1$Left derivative $\ne$ Right derivative$\therefore |x|$ is not differentiable at $x=0$Hence, the correct option is (C).