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 2006-06-25, 16:09 #12 Damian     May 2005 Argentina 2·3·31 Posts 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.
 2006-06-25, 21:59 #13 jinydu     Dec 2003 Hopefully Near M48 33368 Posts 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
 2006-06-26, 15:14 #14 Damian     May 2005 Argentina 2·3·31 Posts 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?
2006-07-01, 22:26   #15
Numbers

Jun 2005
Near Beetlegeuse

22·97 Posts

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

2006-07-01, 22:32   #16
drew

Jun 2005

2·191 Posts

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.

 2006-07-02, 23:48 #17 jinydu     Dec 2003 Hopefully Near M48 2·3·293 Posts Take your favorite piecewiese-continuous (but not continuous) function and integrate it. That should do the trick.
 2006-07-04, 13:06 #18 Damian     May 2005 Argentina 2×3×31 Posts 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

 Similar Threads Thread Thread Starter Forum Replies Last Post Unregistered Homework Help 1 2012-02-09 04:35 Damian Math 0 2006-10-04 13:59

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