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 |
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
equation in functional analysis
ⓘ
equation in real analysis ⓘ functional equation ⓘ mathematical concept ⓘ |
| alsoKnownAs |
Cauchy functional equation
ⓘ
surface form:
additive Cauchy 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.
this entity surface form:
additive Cauchy equation