CameraIcon
CameraIcon
SearchIcon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

An example of a function which is everywhere continuous but fails to be differentiable exactly at two points is ____________.

Open in App
Solution


Consider the function gx=x-1+x+1.

x-1=x-1,x1-x-1,x<1

x+1=x+1,x-1-x+1,x<-1

gx=x-1+x+1=-x-1-x+1,x<-1-x-1+x+1,-1x<1x-1+x+1,x1

gx=x-1+x+1=-2x,x<-12,-1x<12x,x1

When x < −1, g(x) = −2x which being a polynomial function is continuous and differentiable.

When −1 ≤ x < 1, g(x) = 2 which being a constant function is continuous and differentiable.

When x ≥ 1, g(x) = 2x which being a polynomial function is continuous and differentiable.

Let us check the continuity and differentiability of g(x) at x = −1 and x = 1.

At x = −1,

LHL = limx-1-gx=limx-1-2x=-2×-1=2

RHL = limx-1+gx=limx-12=2

g-1=2

Since limx-1-gx=limx-1+gx=g-1, so the function g(x) is continuous at x = −1.

At x = 1,

LHL = limx1-gx=limx12=2

RHL = limx1+gx=limx12x=2×1=2

g1=2×1=2

Since limx1-gx=limx1+gx=g1, so the function g(x) is continuous at x = 1.

Thus, the function g(x) is continuous everywhere i.e. for all x ∈ R.

Now,

Lg'-1=limh0g-1-h-g-1-h

Lg'-1=limh0-2-1-h-2-h

Lg'-1=limh02h-h

Lg'-1=-2

And

Rg'-1=limh0g-1+h-g-1h

Rg'-1=limh02-2-h

Rg'-1=limh00-h

Rg'-1=0

Lg'-1Rg'-1

So, g(x) is not differentiable at x = −1.

Also,

Lg'1=limh0g1-h-g1-h

Lg'1=limh02-2×1-h

Lg'1=limh00-h

Lg'1=0

And

Rg'1=limh0g1+h-g1h

Rg'1=limh021+h-2×1h

Rg'1=limh02hh

Rg'1=2

Lg'1Rg'1

So, g(x) is not differentiable at x = 1.

Thus, the function g(x) is differentiable everywhere except at x = −1 and x = 1.


An example of a function which is everywhere continuous but fails to be differentiable exactly at two points is gx=x-1+x+1 .

flag
Suggest Corrections
thumbs-up
10
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Continuity of Composite Functions
MATHEMATICS
Watch in App
Join BYJU'S Learning Program
CrossIcon