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Old 2006-06-25, 16:09   #12
Damian
 
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Default Differentiable but not C1

In Marsden's Vector Calculus book, it says that a function is called C1 at a point A, if its partial derivatives exist and are continious in the neighborhood of A.
It also says that if a function is C1 at A, then it is differentiable at A.

But that a function been differentiable not necesarily is C1.
Has anybody a counterexample for showing this? I mean, a function been differentiable at a point, but not been C1 there.

Because by reading "upside down" the proof of C1 imply dif, I think it also demostrates that dif implies C1, so C1 and dif are equal.
Thanks.
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Old 2006-06-25, 21:59   #13
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To answer your question Damian, just because a function is differentiable doesn't mean that the derivative is continuous. If I am not mistaken, this is equivalent to saying that there exist discontinuous functions that have an antiderivative.

Last fiddled with by jinydu on 2006-06-25 at 22:04
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Old 2006-06-26, 15:14   #14
Damian
 
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Can you give me an example of a function that is differentiable in the neighborhood of a point, but whose derivative isn't continious there?
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Old 2006-07-01, 22:26   #15
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Quote:
Originally Posted by Damian
Can you give me an example of a function that is differentiable in the neighborhood of a point, but whose derivative isn't continious there?
I should think f(x) = tan(x) would do.

This is easily differentiable, but it's derivative
is not continuous where cos(x) = 0
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Old 2006-07-01, 22:32   #16
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Quote:
Originally Posted by Numbers
I should think f(x) = tan(x) would do.

This is easily differentiable, but it's derivative
is not continuous where cos(x) = 0
I don't think that's what Damian intended, since the point itself is undefined at that value. I think something where the function is defined, but not the derivative is what he was asking.

How about abs(x) at x=0, instead.
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Old 2006-07-02, 23:48   #17
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Take your favorite piecewiese-continuous (but not continuous) function and integrate it. That should do the trick.
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Old 2006-07-04, 13:06   #18
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Thanks for all your answers.
Jinydu, I think your example does the trick.

Anyway I found a good example of a function differentiable but with derivative not continious here:
http://pirate.shu.edu/projects/reals/cont/fp_c1.html
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