GATE CSE 1998 | Question: 1.4

Question

Consider the function $y=|x|$ in the interval $[-1, 1]$. In this interval, the function is

• A. continuous and differentiable
• B. continuous but not differentiable
• C. differentiable but not continuous
• D. neither continuous nor differentiable

Comments

1. Continuous as continuity is maintained as can be seen in graph

 

2. But as graph contains sudden change in direction or break at x = 0 thus not differentiable.

Hence option b is correct.

Answer 1

$f(x) = \begin{cases} -x & \text{ if }  x < 0 \\ +x & \text{ if } x \geq 0  \end{cases}$

Continuity at $ x=0 $

$\begin{align*} \lim_{x \rightarrow 0^{-}}f(x)\ &= \lim_{x \rightarrow 0^{+}}f(x)\\ 0 &= 0\\ \end{align*}$

So $ f(x) = |x|$ is continous at x = 0

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Differentiability at $x=0$

$\begin{align*} \lim_{h \rightarrow 0}\frac{f(0)-f(0-h)}{0-(0-h)} &= \lim_{h \rightarrow 0}\frac{f(0+h)-f(0)}{(0+h)-(0)}\\ \lim_{h \rightarrow 0}\frac{0-(-0+h)}{h)} &= \lim_{h \rightarrow 0}\frac{(0+h)-0}{h} \\ -1 &= 1\\ \end{align*}$

So $ f(x) = |x|$ is not differentiable at x = 0

Option $B$ is the answer

Answer 2

answer is b) continuous but not differentiable  ....because at x=0 this function is not differentiable