Cauchy functional equation
E387476
The Cauchy functional equation is a fundamental equation in functional analysis and real analysis, typically of the form f(x + y) = f(x) + f(y), whose solutions characterize additive functions and illustrate the contrast between regular (e.g., continuous) and highly pathological behaviors.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Cauchy functional equation canonical | 1 |
| additive Cauchy equation | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T3780916 — 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: Cauchy functional equation Context triple: [Ulam stability, appliesTo, Cauchy functional equation]
-
A.
Cauchy–Euler equation
The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
-
B.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
C.
Ulam stability
Ulam stability is a concept in the theory of functional equations that studies when approximate solutions imply the existence of exact solutions nearby, forming the basis of what is now called Hyers–Ulam stability.
-
D.
Cauchy–Riemann equations
The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).
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E.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
- 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: Cauchy functional equation Target entity description: The Cauchy functional equation is a fundamental equation in functional analysis and real analysis, typically of the form f(x + y) = f(x) + f(y), whose solutions characterize additive functions and illustrate the contrast between regular (e.g., continuous) and highly pathological behaviors.
-
A.
Cauchy–Euler equation
The Cauchy–Euler equation is a type of linear ordinary differential equation with variable coefficients that often appears in problems with power-law or scale-invariant behavior.
-
B.
Cauchy–Kovalevskaya theorem
The Cauchy–Kovalevskaya theorem is a fundamental result in partial differential equations that guarantees the existence and uniqueness of analytic solutions to certain initial value problems under appropriate analyticity conditions.
-
C.
Ulam stability
Ulam stability is a concept in the theory of functional equations that studies when approximate solutions imply the existence of exact solutions nearby, forming the basis of what is now called Hyers–Ulam stability.
-
D.
Cauchy–Riemann equations
The Cauchy–Riemann equations are fundamental conditions in complex analysis that characterize when a complex-valued function is holomorphic (complex differentiable).
-
E.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
equation in functional analysis
ⓘ
equation in real analysis ⓘ functional equation ⓘ mathematical concept ⓘ |
| alsoKnownAs |
additive Cauchy equation
ⓘ
linked to:
Cauchy functional equation
|
| appearsIn |
graduate functional analysis courses
ⓘ
undergraduate real analysis courses ⓘ |
| characterizes | additive functions ⓘ |
| codomainTypically | real numbers ⓘ |
| domainTypically | real numbers ⓘ |
| expressesProperty | additivity ⓘ |
| generalCodomain | abelian group ⓘ |
| generalDomain | abelian group ⓘ |
| hasForm | f(x + y) = f(x) + f(y) ⓘ |
| hasPathologicalSolutions |
everywhere discontinuous additive functions
ⓘ
non-measurable additive functions ⓘ |
| hasRegularSolutions | linear functions f(x) = ax ⓘ |
| hasSolutionSpace | vector space over rationals ⓘ |
| hasVariant |
exponential Cauchy equation f(x + y) = f(x)f(y)
ⓘ
multiplicative Cauchy equation f(xy) = f(x)f(y) ⓘ |
| illustrates |
contrast between regular and pathological functions
ⓘ
role of regularity assumptions in analysis ⓘ |
| impliesUnderBoundedOnInterval | f(x) = ax for some constant a ⓘ |
| impliesUnderContinuity | f(x) = ax for some constant a ⓘ |
| impliesUnderMeasurability | f(x) = ax for some constant a ⓘ |
| namedAfter | Augustin-Louis Cauchy ⓘ |
| pathologicalSolutionsDependOn | axiom of choice ⓘ |
| relatedTo |
Hamel basis of R over Q
ⓘ
Jensen inequality ⓘ |
| requiresConditionForRegularity |
boundedness on an interval
ⓘ
continuity at one point ⓘ local boundedness ⓘ measurability ⓘ monotonicity ⓘ |
| solutionDeterminedBy | values on a basis of R as a Q-vector space ⓘ |
| specialCaseOf | Jensen functional equation ⓘ |
| typicalSolutionProperty |
f(-x) = -f(x)
ⓘ
f(0) = 0 ⓘ f(nx) = n f(x) for integer n ⓘ f(qx) = q f(x) for rational q ⓘ |
| usedAs |
standard example in functional equations theory
ⓘ
standard example of non-measurable functions ⓘ |
| usedIn |
functional analysis
ⓘ
group theory ⓘ measure theory ⓘ probability theory ⓘ real analysis ⓘ vector space theory ⓘ |
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: Cauchy functional equation Description of subject: The Cauchy functional equation is a fundamental equation in functional analysis and real analysis, typically of the form f(x + y) = f(x) + f(y), whose solutions characterize additive functions and illustrate the contrast between regular (e.g., continuous) and highly pathological behaviors.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Cauchy functional equation