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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 1:23 PM

On 08/04/2013 10:55 AM, G Patel wrote:
> On Sunday, August 4, 2013 10:48:12 AM UTC-4, 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 something-or-other involving pi. I've
>>
>> forgotten the details--how 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

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