Dear Jean-Claude Evard ,

Differentiable function, It means continuous too, It's derivative is zero (everywhere) it turns that function should be constant, regardless of the domain.

But, if the derivative is zero (almost everywhere), then Cantor function

satisfies this property.

Quoting:

{ f(x) : [0,1] → [0,1], let x be in [0,1] and obtain f(x) by the following steps:

The following is an example for "ENTIRE FUNCTIONS WHOSE DERIVATIVES ARE EQUAL TO ZERO AT p POINTS". obtained by Gel'fond and Ibragimov {A. O. Gel'fond, Selected Works [in Russian], Nauka, Moscow (1973).}

http://booksc.org/book/6360365/0b58ce

As indicated in the article : Vector Valued Analytic Functions by Daniel O. Etter Jr. (see

https://www.ams.org/journals/tran/1965-119-02/S0002-9947-1965-0188750-X/S0002-9947-1965-0188750-X.pdf)

this seems to have been first remarked by Paul Levy in his book :

Processus Stockastiques et Mouvement Brownien, Gauthier-Villars, Paris 1948 page 57.

@ Robert,

It is a good article with new remarkable ideas therein.

Over complex domains, a function may be differentiable at some point but not analytic there, you know it leads to apply Cauchy - Riemann conditions in some neighborhood of that point. The two concepts differentiable and analytic are different if the domain is either real or complex.

Notice that many strange results appeared over complex domains such as

[Liouville's theorem, states that every bounded entire function must be constant. Consequently, its derivative is zero everywhere]

In Jean question, he wonders if such phenomenon, nonconstant function with zero derivative, where the considered variable is real.

[ ...there exists an example of a non-constant differentiable function of a real variable with values... ]

Best regards

How about a whole sequence of domains, with sudden jumps in between, all

totally horizontal, but different heights.?

This makes it nonconstant with zero derivative.

@ Juan,

This function is not differentiable at the jumps.

Example: the function f(x) = [x] the greatest integer function

it designs exactly fit your description.

Unfortunately, it is not differentiable for every integer x and its derivative is zero if x is not an integer.

Dear Issam, I think that the remark of Paul Levy can be also done also for a function of a real variable. If I have such a function for a complex variable, I can choose 2 points where the values of the function are distinct. Now I take the line between these 2 points and I set this line linearly with a real parameter. By composition of functions I think that I obtain a non-constant function of a real variable with a null derivative which is not constant. The point is that the image of the function is in a non-locally convex space.

Am I right?

Dear Robert,

Let us discuss your idea.

We have an analytic non constant function f(z) = u+iv defined over a domain D.,i.e., f(z) : D into f(D) , the derivative of f(z) is zero.

The idea is to restrict f(z) over a line segment L subsets the domain D, where L: Z = Z(t) =x(t) +iy(t), a< t < b.

The new domain is L subset D.

Your claim is

f(z(t)) : L into f(L) , the derivative of f(z(t)) is zero .

Is it?

Discussion:

According to our assumption f(z) is non constant, but it's derivative

df(z)/d(z)= u_x (x,y)+iv_y(x,y) = 0.

Now this should be hold over Z(t) = x(t) +iy(t)

We have u_x (x,y)=0 , v_y(x,y), then using chain rule

u_t (x(t),y(t)) =u_x (x(t),y(t)).dx/dt+u_y (x(t),y(t)).dy/dt=0

v_t (x(t),y(t)) =v_x (x(t),y(t)).dx/dt+v_y (x(t),y(t)).dy/dt=0

u_t (x(t),y(t)) = 0 and v_t (x(t),y(t)) = 0

this leads to u(x(t),y(t))=c constant and v(x(t),y(t))=d constant.

in other words, f(z) restricted to the line segment L is

f(z(t))= c + i d = (c,d) one point which is constant.

Best regards

Dear Issam,

it is not exactly my idea, because you write the image f(z) as a complex number and not z. But the image is in a non-locally convex space. Here I have a function f(z) with complex variable and image in a non-locally convex space (which is non constant and has a null derivative) and two points z_0 and z_1 such that f(z_0) \neq f(z_1). The line throw z_0 and z_1 is given by z=(1-t)z_0+tz_1 where t is a real parameter. I consider the function F(t)=f((1-t)z_0+tz_1)). The derivative of this function F(t) is

f'((1-t)z_0+tz_1)*(z_1-z_0)=0.

And F(0)=f(z_0) , F(1)=f(z_1). Then F'(t)=0 and F(0) \neq F(1).

Best regards

Dear Robert,

The W= f(z) here, considered as a map transforms regions

in R² (xy- plane) into regions in R² (uv -plane).

The parametric equation of a path C ( as a special case z(t)=(1-t)z₀+tz₁ )

is given by F(t)=f(z(t)) , t∈[a,b]. And

dF(t)/dt = df(z(t)/dt .d(z(t))/dt, here

all the terms dF(t)/dt, df(z(t))/dt and d(z(t))/dt represent complex

numbers and not real numbers.

If we want to introduce real variables, we need to introduce

the notion of Jacobian matrix, for f(x,y) = (u(x,y),v(x,y) )as

a mapping from R² into R², its derivative is given by the 2x2 matrix

∂(u,v)/(∂(x,y) = [ ∂u/∂x ∂u/∂y

∂v/∂x ∂v/∂y ]

Best regards

wow that is interesting; bout the closest i can guess is stuff like

D[Cos[t]^2 + Sin[t]^2, t] = 0

@ Valentino Anthony Simpao ,

f(t) = Cos[t]^2 + Sin[t]^2 is a constant function,

where Cos[t]^2 + Sin[t]^2 =1 for any t.

If you mean f(t) = Cos[t^2] + Sin[t^2], then the derivative is not 0.

By definition, an analytic function is locally equal to its Taylor series. Thus, if the derivative is zero, the function is necessarily constant on each connected component of its domain.

But an example of a non constant function with zero derivative is easily provided by functions not defined on a connected domain. It can have any constant value on each connected component of its domain, and consequently derivative zero, and yet be not globally constant. For example, take the sign function defined on R*. It is not constant but its derivative is zero everywhere. Similarly for the floor function, defined on R \ Z.

@ Maximilian,

According to Jean-Claude question, the derivative is zero everywhere.

Yes but I think that the initial question is for a function defined on R (not on a non-connected domain). The important point is that the image of the function is in a topological vector space non-locally convex. The notion of derivative is defined using the limit in the sense of this non-locally convex topology.

I verified yesterday in the old book from Paul Levy : Processus Stockastiques et Mouvements Browniens (the second edition of 1965) and such an example is explained page 57 (and the used function is defined page 26 formula 16). In this example of a function of a real variable, the topology of the image space is the the topology of convergence in probability.

@ Robert,

Good, but my doubt that: is the domain of the constructed real function (everywhere)?

I didn't have access to the text book. Kindly, could you show that formula explicitly, or you can attach some selected pages.

Best wishes

Dear Robert,

I got the link; I will download it as soon as possible.

Then I will take a look at the mentioned formula.

Thank you for caring and your time is highly appreciated.

Best regards

Answer posted on the Web site of Research Gate on April 27, 2018

Victory! This is the happiest day of my life! I will order some champagne! I have just found a reference to a fantastic example of a non-constant function g defined on the set R of real numbers that has derivative zero everywhere.

This function takes its values in a non-locally convex topological vector space E, and even more, it is the restriction g to R of a function f defined on C.

The example is given in following wonderful article, which is of the most hyper-highest quality:

Examples of differentiable mappings into non-locally convex spaces

Helge Glockner

6 pages, Arxiv, July 3 2003)

https://arxiv.org/abs/math/0307040

It is available for free (“Can you believe this?) at the following Web address:

https://arxiv.org/pdf/math/0307040.pdf

The topological vector space E and the function f are defined in Example 1, on page 2.

The definition of the function g, as restriction of the function f, and the properties of this function g are proved in Corollary 1, on page 4.

I thank a lot Issam Kaddoura, Maged Gumaan, Bin-Saad, Robert Rolland, Juan Weisz, Valentino Anthony Simpao, and Maximilian F. Hasler for their answers. These answers are all very interesting.

I am very impressed by the references

Vector Valued Analytic Functions, Daniel O. Etter Jr., and Paul Levy, Processus Stockastiques et Mouvement Brownien, Gauthier-Villars, Paris 1948 page 57. found by Robert Rolland.

I have been very interested in the fantastic “Cantor function” communicated by Issam Kaddoura.

I have been very interested in the reference "ENTIRE FUNCTIONS WHOSE DERIVATIVES ARE EQUAL TO ZERO AT p POINTS". Gel'fond and Ibragimov

{A. O. Gel'fond, Selected Works [in Russian], Nauka, Moscow (1973).}

communicated by Maged Gumaan Bin-Saad.

Issam Kaddoura, Robert Rolland, and Valentino Anthony Simpao were right to think about extensions of the domain from R to C, given that the example in the fantastic article of Helge Glockner is first an example over C, and second it is restricted to R.

Juan Weisz, Maximilian F. Hasler, Issam Kaddoura, and Robert Rolland were right to make me realize that in my question posted on Research Gate, I forgot the hypothesis that the domain is connected.

I will post a new project about non-locally convex spaces on the Web site of Research Gate in a few days.

I hope it will be possible for everyone to find the Web page of this new project by starting from the Web page:

https://www.researchgate.net/profile/Jean_Claude_Evard/contributions

and click on “Projects”.

I have terrible difficulties to navigate on the Web site of Research Gate. I wanted to post a “Thank you” note with comments for every answer, but it is impossible to post comments. We can only post answers.

With best regards,

Jean-Claude

Answer posted on May 19, 2018

I thank Ab. Hamid Ganie a lot for his question about my answer to my question. I am glad that Ab. Hamid is surprised by the fantastic example found by Helge Glockner, and that he is interested in understanding this example. Being surprised and trying to understand is part of being a scientist. I am also surprised by and trying to understand this example as deeply as possible. I am convinced that there is a lot of high value possible research work in this direction, with the possibility to obtain high value results, publications in top journals of mathematics, and become a famous mathematician.

The first steps of such a wonderful research work may be the following:

1. Study meticulously the part of the book of Rodney Coleman whose reference is given in my answer posted on May 2, 2018, as answer to my question “Is there a proof that functions of a real variable with values in a locally convex space and derivative equal to zero are constant?” posted on April 17, 2018 and try to understand every step of Rodney’s proof as deeply as possible.

2. Search the Internet for extensions of Rodney’s proof.

3. If this Internet search is unsuccessful, try to extend Rodney’s proof from normed space to locally convex metric vector spaces.

4. Study meticulously Helge’s article whose reference is given in my answer of April 27, 2018 and try to understand every step very deeply.

5. Search the Internet for other examples of a nonconstant differentiable function defined on an open interval of R, whose derivative is zero everywhere

6. Search the Internet to find all related publications.

While our intuition is essential to find new ideas, Helge’s examples shows that we cannot trust our intuition. This is one of the reasons why I posted my project “Construction of a set of rigorous books of mathematics” on May 15, 2018.

Ab. Hamid’s question encourages me to look for more publications on nonconstant functions whose derivative is zero, and I thank him a lot for this.

With best regards, Jean-Claude

@Issam Kaddoura : in the examples I gave, the derivative IS zero everywhere, but the domain of definition is not a connected space. In this case, the function is not globally constant : it can have a different value on each connected component of its domain of definition. (There is no such thing as a function "defined everywhere". Any function is only defined on its domain and for any such domain, there are always huge spaces which are not in that domain. There are not more points in R = (-oo, oo) than in X = ]0,1[ U ]1,2[. You cannot say a function defined on R is defined everywhere while a function defined on X is not defined everywhere. Both functions are defined exactly on their domain and nowhere else.

OTOH, a function defined on R or on R^n must necessarily be constant if its derivative (defined in the usual way : f(x+h) = f'(x).h + o(h) as h -> 0, for *all* h) is zero everywhere.

Answer posted on May 20, 2018

I thank Peter Breuer and Maximilian Hasler a lot for their comments of yesterday.

The reference that I posted on this Web page, on April 27, 2018, is the following:

Helge Glockner

https://www.researchgate.net/profile/Helge_Glockner

Examples of differentiable mappings into non-locally convex spaces

https://www.researchgate.net/publication/2107743_Examples_of_differentiable_mappings_into_non-locally_convex_spaces

Arxiv, July 3, 2003

https://arxiv.org/abs/math/0307040

Free pdf copy

https://arxiv.org/pdf/math/0307040.pdf

The above article gives an example of a non-constant function g defined on the set R of real numbers that has derivative zero everywhere. This function takes its values in a non-locally convex topological vector space E. It is the restriction g to R of a function f defined on C. The topological vector space E and the function f are defined in Example 1, on page 2. The definition of the function g, as restriction of the function f is on page 4, and the properties of this function g are proved in Corollary 1, on page 4.

===========================

The following reference was given on this Web page, one month ago, by Robert Rolland:

Vector Valued Analytic Functions

Daniel O. Etter Jr.

Transactions of the American Mathematical Society

Volume 119, pages 352—366, 1965

https://www.ams.org/journals/tran/1965-119-02/home.html

https://www.ams.org/journals/tran/1965-119-02/S0002-9947-1965-0188750-X/S0002-9947-1965-0188750-X.pdf

In the introduction of above article, Daniel says the following for any function defined on a nonempty open interval of the set R of real numbers with values in a non-locally convex topological vector space:

1. Such a function can be nonconstant and have derivative equal to zero at every point of the given interval.

2. Such a function can be continuous and fail to be integrable with respect to Lebesgue measure.

In this case the fundamental theorem of calculus is not possible.

======================================

The following reference was also given above, on this Web page, one month ago, by Robert Rolland:

“this seems to have been first remarked by Paul Levy in his book:”

Processus Stockastiques et Mouvement Brownien,

Paul Levy

Gauthier-Villars, Paris 1948 page 57.

24 days ago, Robert Rolland added the following two pieces of information on this Web page:

“I think that the initial question is for a function defined on R (not on a non-connected domain). The important point is that the image of the function is in a topological vector space non-locally convex. The notion of derivative is defined using the limit in the sense of this non-locally convex topology.”

“I verified yesterday in the old book from Paul Levy: Processus Stockastiques et Mouvements Browniens (the second edition of 1965) and such an example is explained page 57 (and the used function is defined page 26 formula 16). In this example of a function of a real variable, the topology of the image space is the topology of convergence in probability.”

============================

I have added a new research project about all this on Research Gate.

============================

With best regards, Jean-Claude

Answer posted on May 20, 2018

I thank Peter Breuer for his comments posted today.

The derivative of a function from an open interval into a topological vector space (TVS) V is defined in the usual way:

https://en.wikipedia.org/wiki/Vector-valued_function#Infinite-dimensional_vector_functions

“Most of the above hold for other topological vector spaces X too.”

The limit is defined in the usual way:

https://en.wikipedia.org/wiki/Topological_space#Continuous_functions

Let g(h) = [f(x + h) – f(x)]/h.

Then a vector A in the TVS V is the limit of g(h) when the nonzero real number h goes to 0 if and only if for every neighborhood W of A, there exists a positive number d, such that for every nonzero real number y in the interval [-d, d], we have that g(y) belongs to the neighborhood W.

Convex sets are well defined in vector spaces:

https://en.wikipedia.org/wiki/Convex_set#In_vector_spaces

I have not changed the setting.

With best regards, Jean-Claude

I have read Glöckners preprint with much interest, it is a very nice construction. However, the second example (Prop.3 and Cor.2) yields a function for which f' = 0 everywhere, but f is not C² ! This means obviously that f'', the second derivative as defined in that article, is not the derivative of f' ! Therefore, I think the adopted notation f^(k) = d^k f / dx^k, is (to the least!) misleading, if not really inappropriate! The notation (d/dx)^k normally by definition means that the operator d/dx is applied k times, but here obviously (d²/dx²)f is NOT (d/dx)(d/dx)f !

Answer posted on May 23, 2018

=====================

I thank a lot Maximilian Hasler and Peter Breuer for their answers posted 3 days ago.

=====================

Below, I talk about the following three articles in chronological order:

=====================

[1] Nancy Lectures on Infinite-Dimensional Lie Groups

Karl-Hermann Neeb

March 12, 2002

https://www.researchgate.net/publication/246917208_Nancy_Lectures_on_Infinite-Dimensional_Lie_Groups

=====================

[2] Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields

Wolfgang Bertram, Helge Glockner, and Karl-Hermann Neeb

March 25, 2003

https://arxiv.org/abs/math/0303300

=========================================

[3] Examples of differentiable mappings into non-locally convex spaces

Helge Glockner

July 3, 2003

https://arxiv.org/abs/math/0307040

=========================================

I use the following abbreviation: [TVS] = [Topological Vector Space]

=========================================

In many publications about TVS, the definition of TVS requires that the underlaying topology of the TVS be Hausdorff. When the Hausdorff condition is not included, we must add it when we talk about limits and derivatives, otherwise limits and derivatives are not unique, and consequently, not well defined.

=========================================

On page 9 of [1], Karl-Hermann gives a definition of derivative for functions from a TVS into a TVS.

=========================================

On page 11 of [1], Karl-Hermann gives an example of a differentiable function from [0, 1] into a TVS whose derivative is zero and that is not constant.

=========================================

On page 2 of [3], Helge gives a sophisticated definition of derivative for functions from a non-discrete topological field into a TVS.

=========================================

On page 4 of [3], in Corollary 1, Helge gives an example of a differentiable function from R into a TVS whose derivative is zero and that is not constant.

=========================================

While my original question is not about higher order derivatives, the observation made by Maximilian about Corollary 2 on page 6 of [3], where the constant function g’ = 0 is not differentiable (g’’ does not exist) gave me heart attack. I agree 100% with everything that Maximilian said about this.

This comes for the sophisticated definition of derivative on page 2. As it is usual in high level research articles, Helge assumes that the readers have read all the articles in the references and the references of the references.

=====================

Concerning continuous functions from R into a TVS, I must split my project “Non-locally convex spaces” into at least two projects: A research project with information about “Continuous on [a, b], not integrable” and another research project with information about “Derivative zero and not constant”.

=====================

I have terrible difficulties with the formatting on Research Gate: Either I get a compact block or I get too many large spaces between the lines.

=====================

With best regards, Jean-Claude

The example given in

Nancy Lectures on Infinite-Dimensional Lie Groups

Karl-Hermann Neeb

March 12, 2002

https://www.researchgate.net/publication/246917208_Nancy_Lectures_on_Infinite-Dimensional_Lie_Groups

is very simple and clear. Here is a screen copy of the example.

Answer posted on May 25, 2018

I thank Robert Rolland and Peter Breuer for their comments of yesterday.

The example given by Robert is the example that I gave above on this page, on May 23, 2018.

It is Example II.2.7 on page 11 of the article [1] of my answer of May 23, 2018, where I said that the example is on page 11 of [1].

I thank Roland for the screen copy of this example. This screen copy saves time to everyone.

Some pieces of information about the spaces l^p in the terrible case 0 < p < 1 are posted on the following two Web pages:

https://en.wikipedia.org/wiki/Lp_space#When_0_

Second answer posted on May 25, 2018

I have just found the following article:

Nonconstant Functions with Derivative Zero: 10772

William C. Waterhouse and Steve Butcher

The American Mathematical Monthly

Volume 107, No 10, page 956

DOI: 10.2307/2695603

December 2000

https://www.researchgate.net/publication/270302876_Nonconstant_Functions_with_Derivative_Zero_10772

http://www.jstor.org/stable/2695603

With best regards, Jean-Claude

Dear Peter,

thank you for your valuable remarks. However, let us specify that the example does not use the derivative of the characteristic function, but the derivative of the vector valued function which to each $t$ associates the characteristic function of the interval $[0,t]$. Then we have a function $F$ from $[0,1]$ into the vector space $L^{1/2}$ (which is a topological vector space where the topology can be given by a distance and not by a norm as remarked by Jean-Claude Evard). The derivative in the example is the derivative of $F$ and not the derivative of an element of $L^{1/2}$ . The fact that the points in $L^{1/2}$ are functions defined almost everywhere is not crucial for the example, because if we do the same thing with the space L^2 where the points are also functions defined almost everywhere, we cannot construct a function $F$ with the required property. The important point is the value of the distance in $L^{1/2}$ as we can see in the computation of the derivative of $F$. Let us remark also that $L^{1/2}$

is a non-locally convex topological vector space .

This leads to the following question: is there such an example for any non-locally topological vector space? Namely, let $V$ a non-locally convex topological vector space, is there a function $F$ from $(0,1)$ to $V$ which is non-constant, derivable and with derivative zero?

I have not seen somewhere an answer to this question.

Best regards,

Robert

Third answer posted on May 25, 2018

I am very grateful to Robert Rolland for his comments, which are a major step in the study of the border between usual and unusual mathematical analysis.

Concerning Robert’s new question, it is a wonderful question. I think that Robert should post it on his RG-Web site to get all the merit and the credit for having found it.

In case that Robert thinks that my suggestion above is good, I also think that it would be better to add in his new question a reference to the discussion on this page, so that everyone can see what references the participants have found so far.

There is no problem for me if Robert does not like this idea. I never forget that Robert was the first to find a reference for my original question, namely the reference to the old book published by Daniel O. Etter Jr. in 1948.

I also thank Robert for the clarifications posted today about the comments from Peter. These clarifications are extremely well-written and make the subject much more accessible to all mathematicians who have never worked in this area.

Now, we have 3 questions about the existence of functions from R into a TVS whose derivative is zero at every point of R and, contrary to usual mathematical analysis, are not constant.

1. My original question on this RG-page:

“Is there at least one example of a such a weird function?”

Updated: “Are there more examples?”

2. My second question posted on another RG-page:

“Is it already proved that such weird functions do not exist if the TVS is locally convex?”

https://www.researchgate.net/post/Is_there_a_proof_that_functions_of_a_real_variable_with_values_in_a_locally_convex_space_and_derivative_equal_to_zero_are_constant

3. Robert’s new question:

“Is it possible to find at least one example for every TVS that is not LC?”

Note: I am currently dividing my old project “Study of Non-LC” into 2 projects: One about derivative zero and another one about not integrable. In these 2 projects, I will write all the credits for all the pieces of information that I have received at the end of these 2 new pages, because it is very distracting to explain difficult mathematics and at the same time give credit. It is like trying to explain how to pilot a helicopter and at the same time give credit to all the scientists who contributed to the creation of helicopters.

I am very grateful for all the comments posted in this discussion.

With best regards,

Jean-Claude

Answer posted on May 29, 2018

=========================

The following article contains a proof that for every complete metric vector space M with trivial dual, there exists a non-constant function f from [0, 1] into M with derivative equal to zero everywhere:

=========================

Curves with zero derivative in F-spaces

N. J. Kalton

Glasgow Mathematical Journal 22(01)

DOI: 10.1017/S0017089500004432

January 1981

https://www.researchgate.net/publication/243031499_Curves_with_zero_derivative_in_F-spaces

https://pdfs.semanticscholar.org/4952/d95d2b8fe8dfe984329415332099684eb3e6.pdf

=========================

It is Theorem 3.3 on page 26.

=========================

In addition, the theorem states that for any elements a and b of M, we can find such a function f such that in addition we have f(0) = a and f(1) = b.

=========================

Corollary 3.4 on page 29 gives a necessary and sufficient condition on b for the existence of such a function f without the hypothesis that M has a trivial dual, and with the hypothesis that a = 0.

=========================

With best regards, Jean-Claude



Answer posted on May 29, 2018

====================

The theorem of Nigel Kalton posted above on this page was conjectured by Stefan Rolewicz in 1972 in the following article:

Stefan Rolewicz

Mémoires de la Société Mathématique de France (1972)

Volume: 31-32, page 327-334

ISSN: 0249-633X

https://eudml.org/doc/94627

http://archive.numdam.org/article/MSMF_1972__31-32__327_0.pdf

====================

This conjecture is Problem 16 on page 333.

====================

With best regards, Jean-Claude