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From: William Elliot on 26 Apr 2010 07:26
On Mon, 26 Apr 2010, Tim Little wrote:
> On 2010-04-26, kelsar777 <kelsar777(a)gmail.com> wrote:
>> On 26 Kwi, 11:46, Maury Barbato <mauriziobarb...(a)aruba.it> wrote:
>>> lim_{h->0} [f(x+h)-f(x-h)] = 0 for every x in R?
>>
>> It's Dirichlet Function: d(x) := 1 for x rational and 0 for x
>> irrational.
>
> That only has the required property for rational x. For every
> irrational x, we can choose arbitrarily small h such that x+h is
> rational. Then x-h is irrational, and therefore f(x+h)-f(x-h) = 1.
>
See my post where I detail that.
From: Dave L. Renfro on 27 Apr 2010 10:09
Maury Barbato wrote:

> Thank you so much, Dave, for your suggestion.
> Your knowledge of mathematical literature goes
> beyond human fancy ... it' really incredible!
> I found a lot of papers on this subject, and also
> the treatise "Symmetric Properties of Real Functions",
> by B. S. Thompson.

In this case it doesn't really count, since just about
anyone alive who is listed in Thomson's book (which I
have, but I didn't have it available when I wrote yesterday's
post) is probably someone I've met or at least have had
"mathematical dealings with". My Ph.D. advisor has written
quite a few papers on the subject matter of Thomson's book,
another student of my advisor who finished the same time
I did wrote his dissertation on this subject, . . .

By the way, Thomson's book makes for a good introduction
to a lot of lesser known hard-core classical real analysis
material, despite it's seemingly very restricted topic. One
reason for this is that many of the results are introduced
and proved without requiring lots of esoteric background
knowledge. Yes, some background knowledge comes up from
time to time, but often those subsections can be skipped
or they can serve as motivation to pursue the background
in other places (most likely cited in his book).

Dave L. Renfro
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