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MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Differentiable

A draft page: Differentiable

Is it correct?

Ideas for improvement?

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Differentiable

Hi;

Looks good!

Interesting is that  the function sin( 1 / x ) is not differentiable at 0 and neither is x sin( 1 / x ) but x^2 sin( 1 / x ) is!

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Nehushtan

Member

Registered: 2013-03-09

Posts: 957

Re: Differentiable

Some corrections.

The Floor and Ceiling Functions are not differentiable, as there is a discontinuity at each jump.

More precisely, they are not differentiable only at integer values of x. (If x is not an integer they are perfectly differentiable at x.)

Similarly the function so y=x[sup](1/3)[/sup] is only not differentiable at the origin; elsewhere it is differentiable.

The y=1/x and  y=sin(1/x) are not defined at the origin so it makes no sense to ask whether they are differentiable there. To be differentiable at a certain point, the function must first of all be defined there!

And the last part:

But a c̶o̶n̶t̶i̶n̶u̶o̶u̶s̶ ̶f̶u̶n̶c̶t̶i̶o̶n̶ function that is continuous at a certain point might not be differentiable at that point, for example the absolute value function is actually continuous (though not differentiable) at the origin.

Last edited by Nehushtan (2013-11-28 22:49:53)

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MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

Re: Differentiable

Thanks bobby and Nehushtan.

May I use some of your wording Nehushtan?

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"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Differentiable

May I use some of your wording Nehushtan?

I certainly would, JFF could not have said it better.

In post #2 I left out the other part of the function definition. I should have said,

f(x) is differentiable at 0 according to the SE. x^2 sin( 1/ x ) would not be because it is not defined at 0.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: Differentiable

hi MathsIsFun

The page looks good to me.  Well done! 

Bob

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Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob