Denjoy–Young–Saks theorem
E850675
The Denjoy–Young–Saks theorem is a result in real analysis that classifies the possible behaviors of the derivative of a real function at almost every point on the real line.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Denjoy–Young–Saks theorem canonical | 2 |
| almost everywhere classification theorem | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T10216096 — 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: Denjoy–Young–Saks theorem Context triple: [Arnaud Denjoy, knownFor, Denjoy–Young–Saks theorem]
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A.
Carathéodory existence theorem
The Carathéodory existence theorem is a result in the theory of ordinary differential equations that guarantees the existence (and sometimes uniqueness) of solutions under weaker regularity conditions on the right-hand side than those required by classical theorems like Picard–Lindelöf.
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B.
Darboux theorem
The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
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C.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
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D.
Banach–Saks theorem
The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
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E.
Du Bois-Reymond function
The Du Bois-Reymond function is a classic example of a continuous but nowhere differentiable function, illustrating pathological behavior in real analysis.
- 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: Denjoy–Young–Saks theorem Target entity description: The Denjoy–Young–Saks theorem is a result in real analysis that classifies the possible behaviors of the derivative of a real function at almost every point on the real line.
-
A.
Carathéodory existence theorem
The Carathéodory existence theorem is a result in the theory of ordinary differential equations that guarantees the existence (and sometimes uniqueness) of solutions under weaker regularity conditions on the right-hand side than those required by classical theorems like Picard–Lindelöf.
-
B.
Darboux theorem
The Darboux theorem is a fundamental result in symplectic geometry stating that all symplectic manifolds are locally symplectomorphic to the standard symplectic space, implying that the symplectic form can always be put into a canonical local normal form.
-
C.
Lebesgue differentiation theorem
The Lebesgue differentiation theorem is a fundamental result in real analysis stating that, for an integrable function, the averages over shrinking neighborhoods converge almost everywhere to the function’s pointwise value.
-
D.
Banach–Saks theorem
The Banach–Saks theorem is a result in functional analysis stating that every bounded sequence in a reflexive Banach space has a subsequence whose Cesàro means converge in norm.
-
E.
Du Bois-Reymond function
The Du Bois-Reymond function is a classic example of a continuous but nowhere differentiable function, illustrating pathological behavior in real analysis.
- F. None of above. chosen
Statements (40)
| Predicate | Object |
|---|---|
| instanceOf | mathematical theorem ⓘ |
| appliesTo | real-valued functions on the real line ⓘ |
| asserts | for any real function, at almost every point, the four Dini derivatives fall into a small number of possible configurations ⓘ |
| assumption | no regularity beyond measurability or being a real function is required ⓘ |
| characterizes | Dini derivatives of real functions almost everywhere ⓘ |
| clarifies | structure of sets where derivatives behave irregularly ⓘ |
| concerns |
almost everywhere properties of derivatives
ⓘ
derivatives of real functions ⓘ pointwise behavior of derivatives ⓘ |
| describes | possible behaviors of the derivative at almost every point ⓘ |
| domain | functions f: \mathbb{R} \to \mathbb{R} ⓘ |
| field |
differentiation theory
ⓘ
measure theory ⓘ real analysis ⓘ |
| guarantees | only finitely many types of derivative behavior at almost every point ⓘ |
| hasConsequence |
constraints on oscillation of real functions at almost every point
ⓘ
description of points of approximate continuity of derivatives ⓘ |
| holds | for Lebesgue almost every point on the real line ⓘ |
| implies |
a real function is approximately differentiable almost everywhere where it has finite Dini derivatives in a certain configuration
ⓘ
classification of differentiability behavior almost everywhere ⓘ |
| language | typically formulated using limsup and liminf of difference quotients ⓘ |
| namedAfter |
Arnaud Denjoy
ⓘ
Stanislaw Saks ⓘ
linked to:
Stanisław Saks
William Henry Young ⓘ |
| relatedTo |
Banach–Zarecki theorem
ⓘ
Lebesgue differentiation theorem ⓘ fine properties of real functions ⓘ functions of bounded variation ⓘ |
| strengthens | basic results on almost everywhere differentiability ⓘ |
| timePeriod | early 20th century ⓘ |
| topic |
classification of one-sided derivatives
ⓘ
pathological behavior of derivatives ⓘ |
| typeOf |
almost everywhere classification theorem
ⓘ
linked to:
Denjoy–Young–Saks theorem
|
| usedIn |
advanced real analysis
ⓘ
fine analysis of singular functions ⓘ geometric measure theory ⓘ |
| usesConcept |
lower left Dini derivative
ⓘ
lower right Dini derivative ⓘ upper left Dini derivative ⓘ upper right Dini derivative ⓘ |
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: Denjoy–Young–Saks theorem Description of subject: The Denjoy–Young–Saks theorem is a result in real analysis that classifies the possible behaviors of the derivative of a real function at almost every point on the real line.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Denjoy–Young–Saks theorem