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Is this the one known as the "blancmage"? Can't remember... Dysprosia 05:32, 27 Jan 2004 (UTC)
Can anyone say why you can't differentiate the function, couldn't you just use the chain-rule?
I think that more accurate plots are needed. Right now there are some obviously incorrect straight lines visible. Pt 21:46, 6 Sep 2004 (UTC)
It's interesting to note that "Kaoseraamat" ("The Chaos Book") by Ülo Lepik and Jüri Engelbrecht (Tallinn 1999; ISBN 9985-50-235-3) gives on page 135 the Weierstrass function as such complex function: where "To visualize it," they take its real part, the Weierstrass-Mandelbrot function: I think these different definitions need further checking. Pt 22:13, 6 Sep 2004 (UTC)
MathWorld gives yet another definition:
Someone should really check some reliable sources — which definition is correct?!
By the way, MathWorld also says that there are actually some points (namely, at for ), where the derivative is finite (). Anywhere else the function is really undifferentiable.
Thus, MathWorld is an interesting source for Wikipedia, but nevertheless needs checking.
-- Pt 22:53, 9 Oct 2004 (UTC)
I moved the sentence here because it certainly needs some discussion before going to the encyclopedia article. -- Pt 23:18, 9 Oct 2004 (UTC)
The definition states: "Almost all continuous functions are nowhere differentiable,..."
But shouldn't it say: "Almost all continuous functions are differentiable."
This looks to be what the MathSource states as well. If you look up the word "differentiable", MathSource states: "Amazingly, there exist continuous functions which are nowhere differentiable." And then lists the Weierstrass and Blancmange functions.
I don't know, maybe i'm talking wacky talk. Sorry if i am.
Sincerely, Mark
Alexandrov and All,
Mark 17:56, 2005 May 20 (UTC)
I just added the requirement that b is an odd integer. This is how I learned it, and is verified by (for instance) [1], [2], [3]. I don't know how to verify its necessity myself, but this requirement is, at least, common. LWizard @ 01:25, 27 April 2006 (UTC)
Would it be possible to get a graphic showing the construction of this function, just to make it clearer? -- 314159 01:05, 27 August 2006 (UTC)
It says in the "elementary" proof that
in the article. But why does a finite sum imply uniform convergence? It seems to me that since, for |b| < 1,
which blows up as , we have that
for all n, thus it is not going to converge to anything as n goes to infinity and so it doesn't look like uniform convergence to me... or did I do something wrong? -- 129.78.64.102 11:39, 18 July 2007 (UTC)
After the proofsketch of the continuity of the function, it says something like
"To prove that it is nowhere differentiable, we consider an arbitrary point x \in {\mathbb R} and show that the function is not differentiable at that point. To do this, we construct two sequences of points xn and x'n which both converge to x, having the property that [...]"
This makes me expect to see more of the proof. The rest is though omitted. Just wondering if something should be added or then change the wording.
In the section "Construction of the Weierstrass function" is:
meant to be:
and is the:
meant to be something else?
RMFan1 (
talk)
15:36, 2 January 2008 (UTC)
No. "inf" and "sup" mean "infimum" and "supremum", respectively.
Ralphmerridew (
talk)
02:46, 13 March 2008 (UTC)
it is described terribly in the text, see Limit superior and limit inferior — Preceding unsigned comment added by 131.111.185.4 ( talk) 00:35, 6 May 2013 (UTC)
Nice picture. But what choices of a and b were made in such plot? —Preceding unsigned comment added by 67.83.187.113 ( talk) 00:44, 30 March 2008 (UTC)
Because f(0)=2, it seems that a=1/2, but b is a bit harder to decipher 130.234.5.138 ( talk) 09:48, 7 October 2010 (UTC)
Experimentally, this seems to be , , which is problematic, since this doesn't meet the condition . -- 67.206.111.10 ( talk) 16:54, 18 January 2022 (UTC)
The discrepancy with the inequality is because the requirement is purely an historic artifact. G. Hardy showed in 1916 that the only true requirement is that .
See for example, * Hardy, G. H. (1916), "Weierstrass's nondifferentiable function" (PDF), Transactions of the American Mathematical Society, 17 (3), American Mathematical Society: 301–325, doi: 10.2307/1989005, JSTOR 1989005
MMmpds 16:59, 19 January 2022 (UTC)
When I try to get the derivative using the differentiation rules, I get this function:
With most values for x, the number changes completely on every iteration of the sum operator. It doesn't converge to a number, which is expectable for a function which hasn't a derivative. But, if you use an integer for x, the argument of the sine function is a multiplycation of pi. In that case the sine function returns 0, which causes the whole function to return 0.
This means the Weierstrass function is differentiable at least some places. These places are where x is an integer, the derivative equals 0 at those places. Do you agree with me?
Paul Breeuwsma ( talk) 23:40, 16 October 2008 (UTC)
b^n will not be an integer in most cases. You COULD still choose x such that xb^n is an integer (e.g. x = m*b^(-n), m an integer), but obviously xb^(n+1) = b*(xb^n) won't be. Thus: choose b a non-integer and the argument for sin will be a non-integer almost always for any x and any n (and for every x, there will be n such that it is non-integer). Also, note the comment from Eric119. In your case, the function could well have a derivative, but it is not proven just by applying a derivative formula for each term. 84.238.115.158 ( talk) 21:11, 16 March 2009 (UTC)
From Hardy's true condition, namely, the requirement that , one finds that the Dini derivatives even at zeros of the function are all infinite. This means the usual derivative cannot exist. So while our minds may wander and confuse us with the lingering thought that the constant function is the only smooth analytic function with compact support, this has no bearing on whether the derivative of a function exists at a zero of the function. In the case of zeros of the function, the derivative does not suddenly become defined. In fact, it suffices to check that none of the Dini derivatives are finite, and therefore the ordinary derivative cannot exist. Since this is true at every point, the function is nowhere differentiable. Another obvious way to see this conclusion is to realize that if one were to translate any W-function up by an amount or down by with , and then ask if one believes that the points formerly crossing the x-axis suddenly gained or lost differentiability, purely due to translation of the graph, the answer is clearly that no such derivative can exist. To make this more logically precise, consider the weak derivative and note that if is a Weierstrass like-function then since both are constants. — Preceding unsigned comment added by MMmpds ( talk • contribs) 17:47, 19 January 2022 (UTC)
The wolfram reference is to a different function by the same name. Probably not a good reference and should be removed. Dannyboytward ( talk) 15:22, 4 November 2008 (UTC)
I've just been reading a book called "Fractals, Chaos, Power Laws", by Manfred Schroeder. It mentions a very interesting property of these functions that I think should be mentioned.
If this function is taken as a timeseries, it's waveform can be played as a musical note. If beta = 2^(13/12) we have (I don't know how to type equations, I appologise profusely)
f(t) = Sum_k[cos(2^(13/12)*t)]
If we double the frequency
f(2*t) = Sum_k[cos(2^(13/12 + 1)*t)] = Sum_k'[cos(2^(13/12)*2^(-1/12)*t)]
if the sum is over the entire audible range, a human perceives
f(2*t) = f(2^(-1/12)*t), or a doubling of frequency results in a paradoxical one semitone decrase in pitch (as opposed to an expected one octave increase)
Dannyboytward ( talk) 15:28, 4 November 2008 (UTC)
The definition given in the article really looks like a Fourier series and in fact, the coefficients would be something like
I think mentioning the connection would improve the article, but i'm a bit hesitant to add it myself since i'm not an expert on the subject.
-Actually never mind, b is not an integer in general. 130.234.5.136 ( talk) 14:19, 6 October 2010 (UTC)
Would it be useful to put in a mention of the distributional derivative of this function? I don't remember seeing a specific reference about it, but it seems that it would be a good example...I imagine that the singular support of the derivative would be equal to the support, but that's just a guess. Anyone have a reference for something like this? -- Spireguy ( talk) 04:16, 16 November 2010 (UTC)
In the article is said that one just proves this:
But I think it is enough to prove:
This is a much weaker condition and therefore probably easier.
I'm of course also interested in the whole proof. Why is the rest omitted? -- Jobu0101 ( talk) 08:58, 18 December 2010 (UTC)
Recently a section was written about the fractional differentiability of the function, and then again removed "as an undue promotion of a result. The edit summary further says: "An equivalent result has been known for many decades already." I'd say the fractional differentiability is quite an interesting property of the function, and a natural generalization of usual differentiability. That is: it is natural and interesting to ask whether a non-differentiable function is fractionally differentiable. The cited article has been cited already 40 times according to Scopus. In short, I think the fractional differentiability deserves at least a sentence or a few in the article. If you disagree, then what is the "equivalent result" known for many decades? This equivalence by itself seems encyclopedia-worthy! — Pt (T) 16:43, 18 December 2010 (UTC)
The last section left me a little confused, as I don't have any background with topology. I was wondering if the last section implied that the set of everywhere differentiable functions on [0,1] is countable. — Preceding unsigned comment added by 2001:18E8:2:1093:41A8:3C62:6615:9534 ( talk) 15:20, 26 November 2012 (UTC)
The external link "Weierstrass function in the complex plane", while pretty, is not a graph of the Riemann-Weierstrass function. — Preceding unsigned comment added by 128.59.193.197 ( talk) 16:18, 23 January 2013 (UTC)
Article says ab must be > 1+3/2*pi = 5.712, but the standard values of a and b, like used in [4], [5], and the example image at the top of the page, are a = 0.5 and b = 3, so ab = 1.5. What is this condition for? — Omegatron ( talk) 15:24, 14 May 2013 (UTC)
The discrepancy with the inequality is because the requirement is purely an historic artifact. G. Hardy showed in 1916 that the only true requirement is that .
See for example, * Hardy, G. H. (1916), "Weierstrass's nondifferentiable function" (PDF), Transactions of the American Mathematical Society, 17 (3), American Mathematical Society: 301–325, doi: 10.2307/1989005, JSTOR 1989005
MMmpds 17:04, 19 January 2022 (UTC)
In the section Construction, the following paragraph appears:
"The computation of the Hausdorff dimension D of the graph of the classical Weierstrass function was an open problem until 2018, while it was generally believed that D = . That D is strictly less than 2 follows from the conditions on and from above. Only after more than 30 years was this proved rigorously."
It is not clear what the word "this" refers to.
Is it only that D < 2 ?
Or is it the full statement that D = ? 2601:200:C000:1A0:C0A5:7B4A:921E:7CA ( talk) 23:29, 5 July 2022 (UTC)
The redirect Nowhere differentiable function has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 May 16 § Nowhere differentiable function until a consensus is reached. Randi Moth Talk Contribs 22:23, 16 May 2023 (UTC)
This
level-5 vital article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||||||||||||||||
|
Is this the one known as the "blancmage"? Can't remember... Dysprosia 05:32, 27 Jan 2004 (UTC)
Can anyone say why you can't differentiate the function, couldn't you just use the chain-rule?
I think that more accurate plots are needed. Right now there are some obviously incorrect straight lines visible. Pt 21:46, 6 Sep 2004 (UTC)
It's interesting to note that "Kaoseraamat" ("The Chaos Book") by Ülo Lepik and Jüri Engelbrecht (Tallinn 1999; ISBN 9985-50-235-3) gives on page 135 the Weierstrass function as such complex function: where "To visualize it," they take its real part, the Weierstrass-Mandelbrot function: I think these different definitions need further checking. Pt 22:13, 6 Sep 2004 (UTC)
MathWorld gives yet another definition:
Someone should really check some reliable sources — which definition is correct?!
By the way, MathWorld also says that there are actually some points (namely, at for ), where the derivative is finite (). Anywhere else the function is really undifferentiable.
Thus, MathWorld is an interesting source for Wikipedia, but nevertheless needs checking.
-- Pt 22:53, 9 Oct 2004 (UTC)
I moved the sentence here because it certainly needs some discussion before going to the encyclopedia article. -- Pt 23:18, 9 Oct 2004 (UTC)
The definition states: "Almost all continuous functions are nowhere differentiable,..."
But shouldn't it say: "Almost all continuous functions are differentiable."
This looks to be what the MathSource states as well. If you look up the word "differentiable", MathSource states: "Amazingly, there exist continuous functions which are nowhere differentiable." And then lists the Weierstrass and Blancmange functions.
I don't know, maybe i'm talking wacky talk. Sorry if i am.
Sincerely, Mark
Alexandrov and All,
Mark 17:56, 2005 May 20 (UTC)
I just added the requirement that b is an odd integer. This is how I learned it, and is verified by (for instance) [1], [2], [3]. I don't know how to verify its necessity myself, but this requirement is, at least, common. LWizard @ 01:25, 27 April 2006 (UTC)
Would it be possible to get a graphic showing the construction of this function, just to make it clearer? -- 314159 01:05, 27 August 2006 (UTC)
It says in the "elementary" proof that
in the article. But why does a finite sum imply uniform convergence? It seems to me that since, for |b| < 1,
which blows up as , we have that
for all n, thus it is not going to converge to anything as n goes to infinity and so it doesn't look like uniform convergence to me... or did I do something wrong? -- 129.78.64.102 11:39, 18 July 2007 (UTC)
After the proofsketch of the continuity of the function, it says something like
"To prove that it is nowhere differentiable, we consider an arbitrary point x \in {\mathbb R} and show that the function is not differentiable at that point. To do this, we construct two sequences of points xn and x'n which both converge to x, having the property that [...]"
This makes me expect to see more of the proof. The rest is though omitted. Just wondering if something should be added or then change the wording.
In the section "Construction of the Weierstrass function" is:
meant to be:
and is the:
meant to be something else?
RMFan1 (
talk)
15:36, 2 January 2008 (UTC)
No. "inf" and "sup" mean "infimum" and "supremum", respectively.
Ralphmerridew (
talk)
02:46, 13 March 2008 (UTC)
it is described terribly in the text, see Limit superior and limit inferior — Preceding unsigned comment added by 131.111.185.4 ( talk) 00:35, 6 May 2013 (UTC)
Nice picture. But what choices of a and b were made in such plot? —Preceding unsigned comment added by 67.83.187.113 ( talk) 00:44, 30 March 2008 (UTC)
Because f(0)=2, it seems that a=1/2, but b is a bit harder to decipher 130.234.5.138 ( talk) 09:48, 7 October 2010 (UTC)
Experimentally, this seems to be , , which is problematic, since this doesn't meet the condition . -- 67.206.111.10 ( talk) 16:54, 18 January 2022 (UTC)
The discrepancy with the inequality is because the requirement is purely an historic artifact. G. Hardy showed in 1916 that the only true requirement is that .
See for example, * Hardy, G. H. (1916), "Weierstrass's nondifferentiable function" (PDF), Transactions of the American Mathematical Society, 17 (3), American Mathematical Society: 301–325, doi: 10.2307/1989005, JSTOR 1989005
MMmpds 16:59, 19 January 2022 (UTC)
When I try to get the derivative using the differentiation rules, I get this function:
With most values for x, the number changes completely on every iteration of the sum operator. It doesn't converge to a number, which is expectable for a function which hasn't a derivative. But, if you use an integer for x, the argument of the sine function is a multiplycation of pi. In that case the sine function returns 0, which causes the whole function to return 0.
This means the Weierstrass function is differentiable at least some places. These places are where x is an integer, the derivative equals 0 at those places. Do you agree with me?
Paul Breeuwsma ( talk) 23:40, 16 October 2008 (UTC)
b^n will not be an integer in most cases. You COULD still choose x such that xb^n is an integer (e.g. x = m*b^(-n), m an integer), but obviously xb^(n+1) = b*(xb^n) won't be. Thus: choose b a non-integer and the argument for sin will be a non-integer almost always for any x and any n (and for every x, there will be n such that it is non-integer). Also, note the comment from Eric119. In your case, the function could well have a derivative, but it is not proven just by applying a derivative formula for each term. 84.238.115.158 ( talk) 21:11, 16 March 2009 (UTC)
From Hardy's true condition, namely, the requirement that , one finds that the Dini derivatives even at zeros of the function are all infinite. This means the usual derivative cannot exist. So while our minds may wander and confuse us with the lingering thought that the constant function is the only smooth analytic function with compact support, this has no bearing on whether the derivative of a function exists at a zero of the function. In the case of zeros of the function, the derivative does not suddenly become defined. In fact, it suffices to check that none of the Dini derivatives are finite, and therefore the ordinary derivative cannot exist. Since this is true at every point, the function is nowhere differentiable. Another obvious way to see this conclusion is to realize that if one were to translate any W-function up by an amount or down by with , and then ask if one believes that the points formerly crossing the x-axis suddenly gained or lost differentiability, purely due to translation of the graph, the answer is clearly that no such derivative can exist. To make this more logically precise, consider the weak derivative and note that if is a Weierstrass like-function then since both are constants. — Preceding unsigned comment added by MMmpds ( talk • contribs) 17:47, 19 January 2022 (UTC)
The wolfram reference is to a different function by the same name. Probably not a good reference and should be removed. Dannyboytward ( talk) 15:22, 4 November 2008 (UTC)
I've just been reading a book called "Fractals, Chaos, Power Laws", by Manfred Schroeder. It mentions a very interesting property of these functions that I think should be mentioned.
If this function is taken as a timeseries, it's waveform can be played as a musical note. If beta = 2^(13/12) we have (I don't know how to type equations, I appologise profusely)
f(t) = Sum_k[cos(2^(13/12)*t)]
If we double the frequency
f(2*t) = Sum_k[cos(2^(13/12 + 1)*t)] = Sum_k'[cos(2^(13/12)*2^(-1/12)*t)]
if the sum is over the entire audible range, a human perceives
f(2*t) = f(2^(-1/12)*t), or a doubling of frequency results in a paradoxical one semitone decrase in pitch (as opposed to an expected one octave increase)
Dannyboytward ( talk) 15:28, 4 November 2008 (UTC)
The definition given in the article really looks like a Fourier series and in fact, the coefficients would be something like
I think mentioning the connection would improve the article, but i'm a bit hesitant to add it myself since i'm not an expert on the subject.
-Actually never mind, b is not an integer in general. 130.234.5.136 ( talk) 14:19, 6 October 2010 (UTC)
Would it be useful to put in a mention of the distributional derivative of this function? I don't remember seeing a specific reference about it, but it seems that it would be a good example...I imagine that the singular support of the derivative would be equal to the support, but that's just a guess. Anyone have a reference for something like this? -- Spireguy ( talk) 04:16, 16 November 2010 (UTC)
In the article is said that one just proves this:
But I think it is enough to prove:
This is a much weaker condition and therefore probably easier.
I'm of course also interested in the whole proof. Why is the rest omitted? -- Jobu0101 ( talk) 08:58, 18 December 2010 (UTC)
Recently a section was written about the fractional differentiability of the function, and then again removed "as an undue promotion of a result. The edit summary further says: "An equivalent result has been known for many decades already." I'd say the fractional differentiability is quite an interesting property of the function, and a natural generalization of usual differentiability. That is: it is natural and interesting to ask whether a non-differentiable function is fractionally differentiable. The cited article has been cited already 40 times according to Scopus. In short, I think the fractional differentiability deserves at least a sentence or a few in the article. If you disagree, then what is the "equivalent result" known for many decades? This equivalence by itself seems encyclopedia-worthy! — Pt (T) 16:43, 18 December 2010 (UTC)
The last section left me a little confused, as I don't have any background with topology. I was wondering if the last section implied that the set of everywhere differentiable functions on [0,1] is countable. — Preceding unsigned comment added by 2001:18E8:2:1093:41A8:3C62:6615:9534 ( talk) 15:20, 26 November 2012 (UTC)
The external link "Weierstrass function in the complex plane", while pretty, is not a graph of the Riemann-Weierstrass function. — Preceding unsigned comment added by 128.59.193.197 ( talk) 16:18, 23 January 2013 (UTC)
Article says ab must be > 1+3/2*pi = 5.712, but the standard values of a and b, like used in [4], [5], and the example image at the top of the page, are a = 0.5 and b = 3, so ab = 1.5. What is this condition for? — Omegatron ( talk) 15:24, 14 May 2013 (UTC)
The discrepancy with the inequality is because the requirement is purely an historic artifact. G. Hardy showed in 1916 that the only true requirement is that .
See for example, * Hardy, G. H. (1916), "Weierstrass's nondifferentiable function" (PDF), Transactions of the American Mathematical Society, 17 (3), American Mathematical Society: 301–325, doi: 10.2307/1989005, JSTOR 1989005
MMmpds 17:04, 19 January 2022 (UTC)
In the section Construction, the following paragraph appears:
"The computation of the Hausdorff dimension D of the graph of the classical Weierstrass function was an open problem until 2018, while it was generally believed that D = . That D is strictly less than 2 follows from the conditions on and from above. Only after more than 30 years was this proved rigorously."
It is not clear what the word "this" refers to.
Is it only that D < 2 ?
Or is it the full statement that D = ? 2601:200:C000:1A0:C0A5:7B4A:921E:7CA ( talk) 23:29, 5 July 2022 (UTC)
The redirect Nowhere differentiable function has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 May 16 § Nowhere differentiable function until a consensus is reached. Randi Moth Talk Contribs 22:23, 16 May 2023 (UTC)