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

$(b)$ $y$ is continuous but not differentiable at $x=0$ as left hand limit will be negative while the right hand limit will be positive but for differentiation, both must be same.

Answer 2

$y = |x| = \begin{Bmatrix} x & x\geqslant 0 \\ -x& x< 0 \end{Bmatrix}$

$y' = \begin{Bmatrix} 1 & x\geqslant 0 \\ -1& x< 0 \end{Bmatrix}$

continuous but not differentiable at x = 0

Answer 3

http://math.stackexchange.com/questions/991475/why-is-the-absolute-value-function-not-differentiable-at-x-0

Answer 4

In simple words, we can say that a function is continuous at a point if we are able to graph it without lifting the pen so it is continous

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

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Differentiable functions are those functions whose derivatives exist. 

If a function is differentiable, then it is continuous. If a function is continuous, then it is not necessarily differentiable.

 

hope my answer helps u a lot