Points of Continuity and Points of Differentiability
Maury Barbato
given a function f:R->R, I set
C(f)={x | f is continuous in x}
D(f)={x | f is differentiable in x}
My questions are the following.
(I) Can we give a characterization of the sets X
such that C(f) = X for some f:R->R?
(II) Can we give a characterization of the sets X
such that D(f) = X for some f:R->R?
These are, surely, hard questions, but maybe there
is some result in the literature (I hope).
Thank you very much for your attention.
My Best Regards,
Maury Barbato
Lars Olsen
<1172839006.43772.12578...@gallium.mathforum.org>,
Maury Barbato <maurizi...@aruba.it> wrote:
A characterization of D(f) is given by the following result due to:
Brudno, A.
Continuity and differentiability. (Russian. English summary)
Rec. Math. [Mat. Sbornik] N.S. 13(55), (1943). 119--134.
Theorem.
Let M be a subset of R. Then the following are equivalent:
(1) There exists a function f;[0,1]\to R such that
M={x|f is differentiable at x}
(2) There exist a G_delta set A and a F_sigma_delta set B with
Lebesgue measure of B = 1
such that
M = A \cap B
This was first proved by Zahorski under the additional assumption that f
had enumerable many points of continuity:
Zahorski, Z. Sur la premi�re d�riv�e. (French) Trans. Amer. Math. Soc. 69,
(1950). 1--54.
and later with out any restrictions by Brudno:
Brudno, A.
Continuity and differentiability. Rec. Math. [Mat. Sbornik] N.S. 13(55),
(1943). 119--134.
For a nice survey paper of these and related topics see:
Bruckner, A. M.; Leonard, J. L. Derivatives. Amer. Math. Monthly 73 1966
no. 4, part II 24--56.
or
Bruckner, Andrew Differentiation of real functions. Second edition. CRM
Monograph Series, 5. American Mathematical Society, Providence, RI, 1994
Maury Barbato
> In article
> <1172839006.43772.1257851018578.JavaMail.root@gallium.
> Zahorski, Z. Sur la première dérivée. (French)
> ) Trans. Amer. Math. Soc. 69,
> (1950). 1--54.
>
> and later with out any restrictions by Brudno:
>
> Brudno, A.
> Continuity and differentiability. Rec. Math. [Mat.
> Sbornik] N.S. 13(55),
> (1943). 119--134.
>
>
> For a nice survey paper of these and related topics
> see:
> Bruckner, A. M.; Leonard, J. L. Derivatives. Amer.
> . Math. Monthly 73 1966
> no. 4, part II 24--56.
>
> or
>
> Bruckner, Andrew Differentiation of real functions.
> . Second edition. CRM
> Monograph Series, 5. American Mathematical Society,
> Providence, RI, 1994
Wauuuu, that's a great result! Two of the papers you
quoted are freely available on the web:
Brudno, A.Continuity and differentiability. Rec. Math.
[Mat. Sbornik] N.S. 13(55),(1943). 119--134.
http://www.mathnet.ru/php/archive.phtml?
wshow=paper&jrnid=sm&paperid=6175&option_lang=eng
Bruckner, A. M.; Leonard, J. L. Derivatives. Amer. Math.
Monthly 73 1966no. 4, part II 24--56.
http://classicalrealanalysis.com/download.aspx
This gives a complete answer to my second question.
Anyhow, I'd be quote surprised if my first question
wouldn't have been answered up until now. All that
I know about the matter is a theorem of V. Volterra,
which is well-known (at least in Italy). You can
find the statement and the proof in the work
Dunham, W. "A Historiacal Gem from Vito Volterra",
http://www.maa.org/mathdl/MM/0025570x.di021166.
02p0055m.pdf
Thank you very much for your help.
Lars Olsen
<1621620773.50741.12579...@gallium.mathforum.org>,
Maury Barbato <maurizi...@aruba.it> wrote:
> Lars Olsen wrote:
>
> > In article
> > <1172839006.43772.1257851018578.JavaMail.root@gallium.
> > mathforum.org>,
> > Maury Barbato <maurizi...@aruba.it> wrote:
> >
> > > Hello,
> > > given a function f:R->R, I set
> > >
> > > C(f)={x | f is continuous in x}
> > >
> > > D(f)={x | f is differentiable in x}
> > >
> > > My questions are the following.
> > >
> > > (I) Can we give a characterization of the sets X
> > > such that C(f) = X for some f:R->R?
> > >
> > > (II) Can we give a characterization of the sets X
> > > such that D(f) = X for some f:R->R?
> > >
> > > These are, surely, hard questions, but maybe there
> > > is some result in the literature (I hope).
> > >
> > > Thank you very much for your attention.
> > > My Best Regards,
> > > Maury Barbato
> >
> >
[Snip]
> Anyhow, I'd be quote surprised if my first question
> wouldn't have been answered up until now. All that
> I know about the matter is a theorem of V. Volterra,
> which is well-known (at least in Italy). You can
> find the statement and the proof in the work
>
> Dunham, W. "A Historiacal Gem from Vito Volterra",
> http://www.maa.org/mathdl/MM/0025570x.di021166.
> 02p0055m.pdf
>
> Thank you very much for your help.
> My Best Regards,
> Maury Barbato
My reason for leaving out the answer to the 1'st question was the
following, namely, that the answer is well-known and a pleasant
exercise in real analysis. Here is the result.
Theorem.
Let M be a subset of R. Then the following are equivalent:
(1) There exists a function f:R\to R such that
M={x|f is continuous at x}
(2) M is a G_delta set.
A proof of this can be found in many textbooks in real analysis, e.g.
Thomson, Bruckner and Bruckner "Elementary real analysis".
Maury Barbato
> In article
> <1621620773.50741.1257941448644.JavaMail.root@gallium.
Ops, I had no idea of that result!
Thank you so much, Lars, for your invaluable help.
Friendly Regards,
Maury Barbato
Dave L. Renfro
>> Theorem.
>> Let M be a subset of R. Then the following are
>> equivalent:
>>
>> (1) There exists a function f:R\to R such that
>>
>> M={x|f is continuous at x}
>>
>> (2) M is a G_delta set.
>>
>> A proof of this can be found in many textbooks in
>> real analysis
Maury Barbato wrote:
> Ops, I had no idea of that result!
> Thank you so much, Lars, for your invaluable help.
I've collected more references than you probably want
to see for this result in the following sci.math post.
See Theorem 2 and Theorem 2' in the post.
References for Continuity Sets [20 December 2006]
http://groups.google.com/group/sci.math/msg/05dbc0ee4c69898e
Dave L. Renfro
Maury Barbato
Thank you very very ... much, Dave! You' re an
extraordinary connoisseur of mathematical analysis and
its history!
Friendly Regards,
Maury Barbato