Reconstruction of a function using Dini derivatives
Dusan
Let f : I ---> R be a function on an open interval I. Assume that the
following two
Dini derivatives D_+ f, D^- f are zero on I. Then f is constant on
I.
Thanks in advance for your help.
Dusan
Dave L. Renfro
> I would like to know if the following assertion holds :
>
> Let f : I ---> R be a function on an open interval I.
> Assume that the following two Dini derivatives
> D_+ f, D^- f are zero on I. Then f is constant on I.
For continuous functions, this is true, but I'm not
sure in general. I'd have to look in some standard
references (Bruckner's "Differentiation of Real Functions"
book, for instance) to see what can happen without
continuity. My guess is that even assuming both of
the Dini derivates you specified are zero in an interval,
we can still have at least one of the other two Dini
derivates be nonzero at each point of a set that is
co-meager (but of measure zero) relative to that interval.
Below are some results under the hypothesis that f
is continuous, lifted from some notes I happen to
have saved on the computer I'm using right now.
Scheeffer [8] (pp. 282-283) proved that we can weaken
"f'=0 at each point" to "f'=0 at co-countably many
points". More precisely, we have the following results,
whose proofs can be found in Rooij/Schikhof [6] (p. 98).
* If f is continuous and f' > 0 at co-countably
many points, then f is strictly increasing on R.
* If f is continuous and f' non-negative at co-countably
many points, then f is non-decreasing on R.
* If f is continuous and f' = 0 at co-countably
many points, then f is constant on R.
See Maurey/Tacchi [4] for a historical essay about
Scheeffer's result. (Their essay also deals with other
results that Scheeffer proved, such as the fact that
given any perfect nowhere dense set P and countable
set Z, there exist a dense set of translations of P
which are disjoint from Z.) More general versions of
Scheeffer's result hold. [[ Replace "f is continuous
at each x" with "the lim-sup (h --> 0+) of f(x-h) is
less than or equal to f(x) is less than or equal to
the lim-sup (h --> 0+) of f(x+h) at each x", and
replace f' with any specified Dini derivate of f. ]]
See Hobson [3] (p. 365), McShane [5] (pp. 200-201),
and Saks [7] (p. 204) for these more general results.
Moreover, we can replace "co-countable" with "complement
of a totally imperfect set" (i.e. f'=0 on a set whose
complement does not contain an uncountable closed set).
See Balaguer [1] [2] and Stocke/Wallin [10]. [[ These
two results are automatic from the fact that the sets
in question are Borel (2'nd level, in fact) and the
fact that "co-countable" is equivalent to "complement
of a totally imperfect set" for Borel sets. ]] The
last three papers also point out that if E does not
have a totally imperfect complement, then there exists
a non-constant continuous function f such that f'=0 on
the complement of E, a result whose proof can also be
found in Semenov [9].
[1] Ferran Sunyer I. Balaguer, "Sur la détermination d'une
fonction par ses nombres dérivés" [On the determination
of a function by its derived numbers], Comptes Rendus
Académie des Sciences (Paris) 245 (1957), 1690-1692.
[MR 19,946a; Zbl 78.04601]
[2] Ferran Sunyer I. Balaguer, "On the determination
of a function by its derived numbers" (Spanish),
Collectanea Mathematica (Barcelona) 10 (1958), 185-194.
[MR 21 #2025; Zbl 85.04501]
[3] Ernest W. Hobson, THE THEORY OF FUNCTIONS OF A REAL
VARIABLE AND THE THEORY OF FOURIER'S SERIES, Volume I,
Dover Publications, 1927/1957, xvi + 732 pages.
[MR 19,1166a; Zbl 81.27702; JFM 53.0226.01]
[4] Bernard Maurey and Jean-Pierre Tacchi, "Ludewig
Scheeffer et les extensions du théorème des
accroissements finis" [Ludwig Scheeffer and the
extensions of the finite-increment theorem],
pp. 1-60 in Travaux Mathématiques XIII, Centre
Universitaire de Luxembourg, 2002.
[MR 2005f:26001]
http://www.math.jussieu.fr/~maurey/articles/
[5] Edward James McShane, INTEGRATION, Princeton University
Press, 1947.
[6] Arnoud C. M. Van Rooij and Wilhelmus H. Schikhof,
A SECOND COURSE ON REAL FUNCTIONS, Cambridge
University Press, 1982, xiii + 200 pages.
[MR 83j:26001; Zbl 474.26001]
[7] Stanislaw Saks, THEORY OF THE INTEGRAL, 2'nd revised
edition, Dover Publications, 1937/1964, xvi + 343 pages.
[Zbl 17.30004; JFM 63.0183.05]
[8] Ludwig Scheeffer, "Zur Theorie der Functionen einer
reellen Veränderlichen" [On the theory of functions
of a real variable], Acta Mathematica 5 (1884),
183-194, 279-296. [JFM 16.0340.01]
http://www.actamathematica.org/fulltext.htm
[9] L. A. Semenov, "On a class of exceptional sets"
(Russian), pp. 132-135 in P. P. Zabreiko (editor),
QUALITATIVE AND APPROXIMATE METHODS FOR THE INVESTIGATION
OF OPERATOR EQUATIONS, Jaroslav. Gos. Univ. (Yaroslavl),
1976. [MR 58 #28348]
[10] Britt-Marie Stocke and Hans Wallin, "Generalizations
of a fundamental theorem in elementary analysis"
(Swedish), Nordisk Matematisk Tidskrift 24 (1976),
33-38. [MR 54 #10521; Zbl 327.26002]
Dave L. Renfro
Michael1
You should take a look at the Denjoy-Young-Saks Theorem.
Hope this helps
Michael
Dave L. Renfro
>> I would like to know if the following assertion holds :
>>
>> Let f : I ---> R be a function on an open interval I.
>> Assume that the following two Dini derivatives
>> D_+ f, D^- f are zero on I. Then f is constant on I.
Dave L. Renfro wrote (in part):
> For continuous functions, this is true, but I'm not
> sure in general. I'd have to look in some standard
> references (Bruckner's "Differentiation of Real Functions"
> book, for instance) to see what can happen without
> continuity. My guess is that even assuming both of
> the Dini derivates you specified are zero in an interval,
> we can still have at least one of the other two Dini
> derivates be nonzero at each point of a set that is
> co-meager (but of measure zero) relative to that interval.
It seems that what I guessed at yesterday is correct.
Given two real numbers b and c with b <= c, there exists
a function f:R --> R such that
(D_+)(f) = b
(D^+)(f) = +infinity
(D_-)(f) = -infinity
(D^-)(f) = c
holds for co-meagerly many x in R (i.e. for all but
a first category set of real numbers x). Choose
b = c = 0 for what you're asking about. The co-meager
set has Lebesgue measure zero, however, by the Denjoy-
Young-Saks theorem. (I'm pretty sure the set can have
Hausdorff dimension 1, though.)
This is Example 3 on p. 205 of
Ludek Zajicek, "On the symmetry of Dini derivates of arbitrary
functions", Comment. Math. Univ. Carolinae 22 (1981), 195-209.
Dave L. Renfro
Dusan
Dave L. Renfro napsal:
Thank you very much for your help !
Best regards
Dusan Bednarik
Dusan
Dave L. Renfro napsal:
But what if I will assume however, that both derivatives are
identically zero in I ? It seems that then f should be continuous on
I. Or it may be even differentiable on I ?
Furthermore, I want to ask another more general question :
suppose that f :I---> R have both Dini derivatives D_- f, D^+f
continuous on I, is it true that then f must be already continuous on
I (or even differentiable on I ? ).
Dusan Bednarik
Dusan Bednarik
Dave L. Renfro
>> Given two real numbers b and c with b <= c, there exists
>> a function f:R --> R such that
>>
>> (D_+)(f) = b
>> (D^+)(f) = +infinity
>>
>> (D_-)(f) = -infinity
>> (D^-)(f) = c
>>
>> holds for co-meagerly many x in R (i.e. for all but
>> a first category set of real numbers x). Choose
>> b = c = 0 for what you're asking about. The co-meager
>> set has Lebesgue measure zero, however, by the Denjoy-
>> Young-Saks theorem. (I'm pretty sure the set can have
>> Hausdorff dimension 1, though.)
>>
>> This is Example 3 on p. 205 of
>>
>> Ludek Zajicek, "On the symmetry of Dini derivates of arbitrary
>> functions", Comment. Math. Univ. Carolinae 22 (1981), 195-209.
Dusan wrote:
> But what if I will assume however, that both derivatives
> are identically zero in I ? It seems that then f should
> be continuous on I. Or it may be even differentiable on I ?
> Furthermore, I want to ask another more general question :
> suppose that f :I---> R have both Dini derivatives
> D_- f, D^+f continuous on I, is it true that then f must
> be already continuous on I (or even differentiable on I ? ).
The example I cited has both Dini derivates zero on the
entire real line. Hence, the Dini two Dini derivates will
be continuous (constant, in fact). Similarly, you can get
both (D_-)(f) and (D^+)(f) equal to zero (hence, continuous) on
the entire real line by a appropriate reflection(s)
(replace x with -x and/or multiply f by -1).
Some of this is on the internet. Off-hand, I don't know
what can happen if we require (D_+)(f) = (D^-)(f) = 0
everywhere and the other two Dini derivates to be
continuous everywhere. One place to look, if it's really
that important to you (and not just idle curiosity) is
the book I mentioned in a post yesterday: "Fine Topology
Methods in Real Analysis and Potential Theory" by
Jaroslav Lukes, Jan Maly, and Ludek Zajicek.
Go to this web page,
http://dz1.gdz-cms.de/no_cache/dms/load/toc/?IDDOC=89486
click on the article that begins on p. 195, then select
p. 205 at the top. This gives the example I posted about.
Go back one page, to p. 204, to see some of the background
constructions related to the example.
Also, go to the google-books search page at
and search with the phrase "Differentiation of Real Functions".
Select the book by Andrew M. Bruckner, which is probably
the top hit, and then click on the chapter title
"The Extreme Der ..." (begins on p. 39). You should be
able to view pp. 39-42.
Also, do a 'search in this book' for "Denjoy-Young"
and select the hit for p. 47. Another version is
on p. 171. (Note: The statement on p. 171 about the
result in Evans/Humke [246] should be "contained in
an F_sigma measure zero set", not "first category set".)
Dave L. Renfro