
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 UTC4, phol...@gmail.com wrote: >> On Saturday, August 3, 2013 1:37:09 PM UTC7, 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 online 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/twinprimes/all/ > .
david
 abc?

