|
|
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.
>>
>>>
>>
>>>
>>
>>>
>>
>>> 2) Do you know a function that is differentiable at an isolated point?
>>
>>>
>>
>>>
>>
>>>
>>
>>> 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
--
abc?
|
|
