{ "model": { "rank": 4915, "code": "thing-q170058", "model_id": "vr.tr.continuous-function", "name": "continuous function", "purpose": "Let an agent explain continuity by definition, tests, theorems and examples for learners and practitioners.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QLT", "industry": "", "version": "", "url": "/models/thing/q170058/", "tier": 2, "score": 71, "payload": { "layer": "wikidata", "aliases": [ "inverse hyperbolic function", "continuous extension", "homeomorphism", "differentiable function", "injective continuous map", "surjective continuous map", "real-valued continuous function of a real variable", "continuous function with compact support", "continuous extremum constrained function", "sublinear function", "convex functional", "inverse hyperbolic tangent and inverse hyperbolic cotangent", "inverse hyperbolic sine and inverse hyperbolic cosine", "power function", "single-variable quadratic function", "elementary function", "local homeomorphism", "proper map", "continuous linear operator", "pseudogamma function", "geometrical continuity", "quasi-fibration", "continuous group action", "weak homotopy equivalence", "singular function", "convex function", "continuous real function having an interval for domain", "bundle map", "Gauss map", "homotopy", "uniformly continuous function", "cellular map", "diffeomorphism", "expansive homeomorphism", "k times continuously differentiable function", "submersion", "immersion", "continuously differentiable function", "topological homomorphism", "proper group action" ], "aliasCount": 55, "merged": 56, "knownIn": 71, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q170058", "url": "https://www.wikidata.org/wiki/Q170058", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T13:41:12Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q170058", "registryId": "vr.tr.continuous-function", "name": "continuous function", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QLT" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q170058/", "model": { "registry_id": "vr.tr.continuous-function", "name": "continuous function", "purpose": "Let an agent explain continuity by definition, tests, theorems and examples for learners and practitioners.", "definition": "A function whose output changes without abrupt jumps, formally one for which small changes in the input produce arbitrarily small changes in the output, as captured by the epsilon-delta definition; continuous functions satisfy results such as the intermediate value theorem, and continuity generalises to topological spaces where homeomorphisms are continuous bijections with continuous inverses.", "what_it_is_for": "Analysis, topology and modelling.", "affordances": [ "test whether a function is continuous", "explain the epsilon-delta definition", "apply the intermediate and extreme value theorems", "relate continuity and differentiability" ], "distinguishing_features": [ "No jumps or breaks", "Defined by limits", "Preserved by composition", "Generalises to topology" ], "appearance": "Not physical; graphs drawn without lifting the pen, and formulas.", "visual_identification": [ "Graphs without jumps or holes", "Epsilon-delta statements", "Step functions are discontinuous" ], "physical_properties": [], "families_and_kinds": [ "continuous real functions", "uniformly continuous functions", "Lipschitz continuous functions", "continuous maps between spaces", "homeomorphisms" ], "related_models": [ { "relation": "is a kind of", "target": "function", "why": "category" }, { "relation": "is required by", "target": "intermediate value theorem", "why": "theorem" }, { "relation": "is weaker than", "target": "differentiability", "why": "related property" }, { "relation": "is related to", "target": "polynomial", "why": "example" } ], "identifiers": [], "standards_and_regulation": [ "No specific regulation" ], "failure_modes_and_hazards": [ "Confusing continuity with differentiability", "Ignoring the domain" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "definition", "name": "Definition", "description": "What continuity means.", "rationale": "Definitions must be exact.", "layers": [ { "id": "epsilon", "name": "Epsilon-delta", "description": "Formal definition.", "findings": [ { "id": "epsilon-finding", "name": "Epsilon-delta", "description": "Formal definition.", "questions": [ { "text": "How does the epsilon-delta definition express continuity at a point?", "kind": "definition" }, { "text": "How is it applied to this function?", "kind": "action" } ] } ] }, { "id": "test", "name": "Test", "description": "Checking.", "findings": [ { "id": "test-finding", "name": "Test", "description": "Checking continuity.", "questions": [ { "text": "Is this function continuous on its domain?", "kind": "boundary" }, { "text": "Where are its discontinuities, and of what type?", "kind": "definition" } ] } ] } ] }, { "id": "theorems", "name": "Theorems", "description": "Key results.", "rationale": "Continuity guarantees results.", "layers": [ { "id": "ivt", "name": "IVT", "description": "Intermediate values.", "findings": [ { "id": "ivt-finding", "name": "IVT", "description": "Intermediate value theorem.", "questions": [ { "text": "How does the intermediate value theorem guarantee a root here?", "kind": "definition" }, { "text": "What conditions must hold?", "kind": "boundary" } ] } ] }, { "id": "evt", "name": "EVT", "description": "Extreme values.", "findings": [ { "id": "evt-finding", "name": "EVT", "description": "Extreme value theorem.", "questions": [ { "text": "Why does a continuous function on a closed interval attain a maximum and minimum?", "kind": "definition" }, { "text": "What fails on open intervals?", "kind": "boundary" } ] } ] } ] }, { "id": "variants", "name": "Variants", "description": "Stronger notions.", "rationale": "Notions differ.", "layers": [ { "id": "uniform", "name": "Uniform", "description": "Uniform continuity.", "findings": [ { "id": "uniform-finding", "name": "Uniform", "description": "Uniform continuity.", "questions": [ { "text": "Is the function uniformly or Lipschitz continuous?", "kind": "definition" }, { "text": "Why does it matter?", "kind": "definition" } ] } ] }, { "id": "topology", "name": "Topology", "description": "General spaces.", "findings": [ { "id": "topology-finding", "name": "Topology", "description": "Topological continuity.", "questions": [ { "text": "How is continuity defined using open sets?", "kind": "definition" }, { "text": "What is a homeomorphism?", "kind": "definition" } ] } ] } ] }, { "id": "learning", "name": "Learning", "description": "Teaching.", "rationale": "Examples help.", "layers": [ { "id": "examples", "name": "Examples", "description": "Worked examples.", "findings": [ { "id": "examples-finding", "name": "Examples", "description": "Examples.", "questions": [ { "text": "Which examples show continuous but not differentiable functions?", "kind": "definition" }, { "text": "How can they be graphed?", "kind": "action" } ] } ] }, { "id": "misconceptions", "name": "Misconceptions", "description": "Common errors.", "findings": [ { "id": "misconceptions-finding", "name": "Misconceptions", "description": "Misconceptions.", "questions": [ { "text": "Which misconceptions about continuity are common?", "kind": "provenance" }, { "text": "How can they be corrected?", "kind": "action" } ] } ] } ] } ] }, "openQuestions": [ "Should kinds of continuity be separate entries?", "How should proofs be linked?", "How should topological continuity be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "abstract concept", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a continuous function, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent explain continuity by definition, tests, theorems and examples for learners and practitioners.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: inverse hyperbolic function, continuous extension, homeomorphism, differentiable function, injective continuous map, surjective continuous map, real-valued continuous function of a real variable, continuous function with compact support, continuous extremum constrained function, sublinear function, convex functional, inverse hyperbolic tangent and inverse hyperbolic cotangent.", "56 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 71 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a continuous function, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see. What is recognised is an instance of it, and which instances count is exactly what is argued about." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "inverse hyperbolic function", "continuous extension", "homeomorphism", "differentiable function", "injective continuous map", "surjective continuous map", "real-valued continuous function of a real variable", "continuous function with compact support", "continuous extremum constrained function", "sublinear function", "convex functional", "inverse hyperbolic tangent and inverse hyperbolic cotangent" ] } ], "standing": "Described in 71 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 56 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one continuous function from another, and a continuous function from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a continuous function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a continuous function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a continuous function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a continuous function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a continuous function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a continuous function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a continuous function sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a continuous function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a continuous function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a continuous function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a continuous function from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a continuous function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a continuous function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a continuous function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a continuous function passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a continuous function between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a continuous function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a continuous function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a continuous function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a continuous function, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a continuous function is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a continuous function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a continuous function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a continuous function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definitions-in-contest", "name": "Definitions and who holds them", "description": "What continuous function is taken to mean, and by whom.", "rationale": "When a field disagrees about a concept, the disagreement is the content. A model that picks one definition silently destroys the information.", "layers": [ { "id": "competing-definitions", "name": "Competing definitions", "description": "The main readings and the traditions behind them.", "findings": [ { "id": "definition-map", "name": "Definitions and their holders", "description": "Each definition with the school or body that holds it.", "questions": [ { "id": "definition-map-q01", "text": "Which definitions of continuous function are in use, and which tradition or body holds each?", "kind": "definition" }, { "id": "definition-map-q02", "text": "What turns on the difference between them in practice?", "kind": "boundary" } ] } ] }, { "id": "operationalisation", "name": "Operationalisation", "description": "How it is measured or applied when it has to be.", "findings": [ { "id": "operational-record", "name": "Measures and proxies", "description": "Instruments and indicators used to stand in for it.", "questions": [ { "id": "operational-record-q01", "text": "How is continuous function operationalised or measured in practice, and by what instrument?", "kind": "measurement" }, { "id": "operational-record-q02", "text": "What does that operationalisation leave out, and when does that matter?", "kind": "boundary" } ] } ] } ] }, { "id": "instances-and-use", "name": "Instances, use and consequence", "description": "What counts as an instance of continuous function and what follows from calling something that.", "rationale": "Applying a concept is an act with consequences, so a model must say what the label licenses and what it does not.", "layers": [ { "id": "instances", "name": "What counts as an instance", "description": "Clear cases, borderline cases and non-cases.", "findings": [ { "id": "instance-tests", "name": "Tests for an instance", "description": "What would settle whether something falls under it.", "questions": [ { "id": "instance-tests-q01", "text": "What would settle whether something is an instance of continuous function?", "kind": "boundary" }, { "id": "instance-tests-q02", "text": "Which borderline cases are argued about, and on what grounds?", "kind": "definition" } ] } ] }, { "id": "consequence", "name": "Consequence of application", "description": "Rights, duties or decisions that follow from the label.", "findings": [ { "id": "consequence-record", "name": "What the label licenses", "description": "What an agent may do once something is classified this way.", "questions": [ { "id": "consequence-record-q01", "text": "What follows practically once something is treated as continuous function?", "kind": "action" }, { "id": "consequence-record-q02", "text": "What must an agent not infer from the label alone?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a continuous function came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a continuous function and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a continuous function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a continuous function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a continuous function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a continuous function is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a continuous function, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a continuous function, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a continuous function is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 5, "layers": 10, "findings": 12, "questions": 32 } } } }