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Topic: Some derivative/continuity questions
Replies: 15   Last Post: Aug 5, 2013 9:47 AM

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 David Bernier Posts: 3,892 Registered: 12/13/04
Re: Some derivative/continuity questions
Posted: Aug 4, 2013 1:04 PM

On 08/04/2013 10:54 AM, G Patel wrote:
> On Saturday, August 3, 2013 6:51:47 PM UTC-4, phol...@gmail.com wrote:
>> On Saturday, August 3, 2013 1:37:09 PM UTC-7, G Patel wrote:
>>

>>> 1) If f is differentiable at x, is it continuous on an interval about x?
>>
>>>
>>
>>> It seems intuitively like it should be.
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>>>
>>
>>>
>>
>>>
>>
>>> 2) Do you know a function that is differentiable at an isolated point?
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>>>
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>>>
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>>>
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>>> Thanks
>>
>>
>>
>> Well, what is normally taught in Calc I is the relationship between a limit, continuity and differentiability.
>>
>> A function is differentiable at a point if it is .....
>>
>> 1/ Defined at that point in the domain of the function.
>>
>> 2/ Continuous at that point (both sides, left and right). This precludes the end points of a line segment.
>>
>> 3/ The left determined derivative equals the right derivative (using the limit definition of a derivative). This precludes a corner point or cusp.
>>
>>
>>
>> So if you apply those requirements to an isolated point.......?
>>
>> Phil H

>
> You folks are on another level.
>

The other day, I felt mathematically inadequate after struggling in
an on-line seminar on Yitang Zhang's recent proof of "bounded gaps"
for the primes.

Yitang Zhang showed that, for infinitely many positive integers n,

| p_{n+1} - p_n | < C,
for C ~= 70,000,000 and where p_n is the n'th prime.

So, p_1 = 2, p_2 = 3 , p_3 = 5 and so on.

From Wired magazine:
< http://www.wired.com/wiredscience/2013/05/twin-primes/all/ > .

david

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