epsilon–delta definition of limit
E110604
The epsilon–delta definition of limit is the rigorous formalization of the intuitive notion of a function approaching a value, forming the foundation of modern analysis and calculus.
All labels observed (2)
| Label | Occurrences |
|---|---|
| epsilon–delta definition of limit canonical | 1 |
| method of limits | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T940257 — 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: epsilon–delta definition of limit Context triple: [Karl Weierstrass, notableFor, epsilon–delta definition of limit]
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A.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
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B.
Riemann sums
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
-
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.
inverse function theorem
The inverse function theorem is a fundamental result in calculus and differential geometry that gives conditions under which a differentiable function has a locally defined differentiable inverse near a point where its derivative is invertible.
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E.
local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
- 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: epsilon–delta definition of limit Target entity description: The epsilon–delta definition of limit is the rigorous formalization of the intuitive notion of a function approaching a value, forming the foundation of modern analysis and calculus.
-
A.
Riemann–Lebesgue lemma
The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis stating that the Fourier coefficients (or transform) of an integrable function vanish at infinity.
-
B.
Riemann sums
Riemann sums are a fundamental method in calculus for approximating the area under a curve by summing the areas of a sequence of rectangles, forming the basis of the definition of the definite integral.
-
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.
inverse function theorem
The inverse function theorem is a fundamental result in calculus and differential geometry that gives conditions under which a differentiable function has a locally defined differentiable inverse near a point where its derivative is invertible.
-
E.
local existence and uniqueness theorem
The local existence and uniqueness theorem is a fundamental result in differential equations that guarantees, under suitable conditions, a single solution passing through a given initial point, valid in some neighborhood of that point.
- F. None of above. chosen
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
concept in real analysis
ⓘ
foundational concept in calculus ⓘ mathematical definition ⓘ |
| alternativeFormulation |
metric-space definition using distances
ⓘ
neighborhood-based definition of limit ⓘ |
| appliesTo |
complex-valued functions
ⓘ
multivariable functions ⓘ real-valued functions of a real variable ⓘ |
| assumes | ordered field structure of real numbers ⓘ |
| characterizes | behavior of functions near a point ⓘ |
| clarifies |
concept of continuity at a point
ⓘ
difference between limit and function value ⓘ |
| compatibleWith | completeness of the real numbers ⓘ |
| coreQuantifiers | for every ε > 0 there exists δ > 0 ⓘ |
| defines | limit of a function at a point ⓘ |
| doesNotRequire | function to be defined at the limit point ⓘ |
| enables |
epsilon–delta proofs of differentiability
ⓘ
rigorous error estimates in analysis ⓘ |
| field |
calculus
ⓘ
mathematical analysis ⓘ |
| formalizes |
idea of a function approaching a value
ⓘ
intuitive notion of limit ⓘ |
| generalizesTo |
metric spaces
ⓘ
topological spaces ⓘ |
| historicallyAssociatedWith |
Augustin-Louis Cauchy
ⓘ
Karl Weierstrass ⓘ |
| implies | uniqueness of limits when they exist ⓘ |
| logicalStructure | universal–existential quantifier pattern ⓘ |
| purpose |
to provide rigorous foundations for calculus
ⓘ
to remove ambiguity from infinitesimal reasoning ⓘ |
| relatedConcept |
epsilon–N definition of limit of a sequence
ⓘ
sequential definition of limit ⓘ |
| requires |
absolute value inequalities
ⓘ
notion of distance on the real line ⓘ |
| supports |
rigorous definition of continuity
ⓘ
rigorous definition of definite integral ⓘ rigorous definition of derivative ⓘ rigorous definition of series convergence ⓘ |
| taughtIn |
introductory real analysis textbooks
ⓘ
undergraduate analysis courses ⓘ |
| timePeriod | 19th century ⓘ |
| typicalFormulation | for every ε > 0 there exists δ > 0 such that 0 < |x − a| < δ implies |f(x) − L| < ε ⓘ |
| usedIn |
epsilon–N definition of sequence limits
ⓘ
proofs of continuity properties ⓘ proofs of limit laws ⓘ |
| usesSymbols |
delta (δ)
ⓘ
epsilon (ε) ⓘ |
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: epsilon–delta definition of limit Description of subject: The epsilon–delta definition of limit is the rigorous formalization of the intuitive notion of a function approaching a value, forming the foundation of modern analysis and calculus.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: epsilon–delta definition of limit