Re: Ullrich the Mathematician!

From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 03/10/05 

Date: Thu, 10 Mar 2005 08:23:54 -0600

On Wed, 09 Mar 2005 10:34:11 -0800, The World Wide Wade
<waderameyxiii@comcast.remove13.net> wrote:

>In article <036u211te4nsq8n69a319th8pd721hko1i@4ax.com>,
> David C. Ullrich <ullrich@math.okstate.edu> wrote:
>
>> On Wed, 09 Mar 2005 07:20:07 -0600, David C. Ullrich
>> <ullrich@math.okstate.edu> wrote:
>>
>> >On 8 Mar 2005 09:43:54 -0800, "Jason" <logamath@yahoo.com> wrote:
>> >
>> >>Ullrich wrote:
>> >>
>> >>> Suppose that f is differentiable on [0,1] (or differentiable
>> >>> on (0,1) and continuous on [0,1]). Does it follow that
>> >>
>> >>> f(1) - f(0) = lim_N sum_1^N f'(j/N)/N ?
>> >>
>> >>> Of course this is clear from the fundamental theorem of calculus
>> >>> if f' is continuous. It seems unlikely that it's true under the
>> >>> stated hypotheses, but "seems unlikely" is not quite a
>> >>counterexample.
>> >>
>> >>Now you talk about my posts being funny! Let's see how funny yours is:
>> >>
>> >>First you say "suppose f is differentiable on [0,1]" and in parentheses
>> >>you write "or differentiable on (0,1) and continuous on [0,1]".
>> >>May I remind you the two are not the *equivalent*. Your statement seems
>> >>to imply they are.
>> >
>> >No, I didn't imply that they are equivalent, I was giving two
>> >possible versions of the theorem.
>> >
>> >(Why? Because the version that most closely matches Gabriel's
>> >statement is much less likely to be true than the other.)
>>
>> And in fact it turns out that the version he states,
>> not assuming differentiability at one of the endpoints,
>> is definitely _false_. This one's an easy example:
>
>[snip of example]:
>
>This is perhaps simpler, and has the virtue of giving
>differentiability everywhere on [0,1]: [..]
>
>So Gabriel fails rather badly here, even though f is
>differentiable everywhere on [0,1]. The problem comes from the
>wildness of f'. A natural remedy is to assume f' is continuous on
>[0,1], but as noted a billion times already, the result is then a
>trivial consequence of FTC. In summary, when Gabriel is of
>interest, it's false. When it's true, it's of no interest.

Yes, that is probably simpler.

Had you pointed out an example like this previously?
(Not worried about copyright or anything, just wondering
whether I missed something - I've been telling people that
the theorem seems pretty likely false but that nobody'd
given an actual counterexample.)

************************

David C. Ullrich