Weierstrass function
E110606
continuous function
example in mathematical analysis
mathematical function
nowhere differentiable function
pathological function
real-valued function
The Weierstrass function is a classic example in mathematical analysis of a continuous function that is nowhere differentiable, illustrating the counterintuitive behavior possible in real-valued functions.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Weierstrass function canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T940259 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Weierstrass function Context triple: [Karl Weierstrass, notableFor, Weierstrass function]
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A.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
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B.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
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C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
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D.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
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E.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Weierstrass function Target entity description: The Weierstrass function is a classic example in mathematical analysis of a continuous function that is nowhere differentiable, illustrating the counterintuitive behavior possible in real-valued functions.
-
A.
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe
Über die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe is Bernhard Riemann’s seminal 1854 paper that laid foundational ideas for Fourier series and modern real analysis, including the concept now known as the Riemann integral.
-
B.
Riemann zeta function
The Riemann zeta function is a complex-valued function central to analytic number theory, whose properties—especially the distribution of its zeros—are deeply connected to the distribution of prime numbers.
-
C.
Riemann–Siegel formula
The Riemann–Siegel formula is an asymptotic expression that efficiently approximates the Riemann zeta function on the critical line, playing a key role in the numerical study of its zeros.
-
D.
Gauss’s constant
Gauss’s constant is a mathematical constant arising in number theory and complex analysis, particularly in connection with the lemniscate and elliptic functions.
-
E.
Riemann–Liouville integral
The Riemann–Liouville integral is a fundamental operator in fractional calculus that generalizes the concept of an n-fold repeated integral to non-integer (fractional) orders.
- F. None of above. chosen
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
continuous function
ⓘ
example in mathematical analysis ⓘ mathematical function ⓘ nowhere differentiable function ⓘ pathological function ⓘ real-valued function ⓘ |
| appearsIn |
classical textbooks on real analysis
ⓘ
literature on fractal geometry ⓘ |
| category |
counterexamples in analysis
ⓘ
special functions in real analysis ⓘ |
| codomain | real numbers ⓘ |
| constructionMethod |
Fourier-like trigonometric series
ⓘ
infinite series ⓘ |
| definedOn | real line ⓘ |
| field |
fractal geometry
ⓘ
functional analysis ⓘ mathematical analysis ⓘ real analysis ⓘ |
| graphProperty |
nowhere differentiable curve
ⓘ
self-similar structure at different scales ⓘ |
| hasRegularity | Hölder continuous of some exponent <1 ⓘ |
| influenced |
development of fractal theory
ⓘ
study of irregular functions ⓘ |
| introducedBy | Karl Weierstrass ⓘ |
| namedAfter | Karl Weierstrass ⓘ |
| parameterCondition |
0<a<1
ⓘ
ab>1+3π/2 ⓘ b is an odd integer ⓘ |
| property |
bounded
ⓘ
everywhere continuous ⓘ fractal graph ⓘ graph has non-integer Hausdorff dimension ⓘ nowhere differentiable ⓘ uniformly continuous on R ⓘ |
| role |
counterexample in analysis
ⓘ
example of continuous nowhere differentiable function ⓘ example showing limits of geometric intuition ⓘ |
| typicalForm | W(x)=∑_{n=0}^{∞} a^n cos(b^n π x) ⓘ |
| usedIn |
studying Hölder continuity
ⓘ
studying function regularity ⓘ teaching real analysis ⓘ |
| yearProposed | 1872 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Weierstrass function Description of subject: The Weierstrass function is a classic example in mathematical analysis of a continuous function that is nowhere differentiable, illustrating the counterintuitive behavior possible in real-valued functions.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.