
Re: Some derivative/continuity questions
Posted:
Aug 4, 2013 1:23 PM


On 08/04/2013 10:55 AM, G Patel wrote: > On Sunday, August 4, 2013 10:48:12 AM UTC4, Peter Percival wrote: >> dullrich@sprynet.com wrote: >> >>> On Sat, 03 Aug 2013 22:13:45 +0100, Peter Percival >> >>> <peterxpercival@hotmail.com> wrote: >> >>> >> >>>> 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 >> >>>> >> >>>> 1) The function whose value is 1/q at rational points p/q (in lowest >> >>>> terms) and whose value is 0 at irrational points, is differentiable at >> >>>> every irrational point, >> >>> >> >>> Who told you that? It's not so. >> >>> >> >>>> but discontinuous at every rational point. >> >>>> >> >>>> 2) The Weierstrass function is f(x) = sum_{n=0}^infty a^n cos(b^n pi x). >> >>>> Consider f(x)x^2, this is differentiable at 0 only. >> >>> >> >>> Yes or no, depending on the value of a and b. >> >> >> >> Oh dear, I forgot to mention that b is a positive odd integer, a is in >> >> (0,1) and ab is greater than somethingorother involving pi. I've >> >> forgotten the detailshow embarrassing. > > What kind of calculus book did you read to learn about these exotic examples? >
It's too advanced (in my opinion) to show the proof in a calculus text.
Maybe "Counterexamples in Analysis", 1964 has it ...
< http://en.wikipedia.org/wiki/Weierstrass_function > .
david
 abc?

