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Research draft

continuous function

vr.tr.continuous-function · XCT.QLT

Let an agent explain continuity by definition, tests, theorems and examples for learners and practitioners.

Thing Registry Cross-cutting context

Research draft, second pass

A second pass drafted this model: the structure a model of this thing needs, and what is known about it in the world. The line under this one says how the second half was obtained - researched against sources, or recalled without web access, in which case nothing here was read anywhere and every claim is a lead to verify. Unreviewed either way.

written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify

Researched by: Claude

Purpose and description

Let an agent explain continuity by definition, tests, theorems and examples for learners and practitioners.

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.

It can be 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

What it looks like

Not physical; graphs drawn without lifting the pen, and formulas.

How it is recognised

Graphs without jumps or holes

Epsilon-delta statements

Step functions are discontinuous

Related models

is a kind of - category

function

is required by - theorem

intermediate value theorem

is weaker than - related property

differentiability

is related to - example

polynomial

In practice

Families and kinds

continuous real functions

uniformly continuous functions

Lipschitz continuous functions

continuous maps between spaces

homeomorphisms

Standards and regulation

No specific regulation

Failure modes and hazards

Confusing continuity with differentiability

Ignoring the domain

Also called

inverse hyperbolic functioncontinuous extensionhomeomorphismdifferentiable functioninjective continuous mapsurjective continuous mapreal-valued continuous function of a real variablecontinuous function with compact supportcontinuous extremum constrained functionsublinear functionconvex functionalinverse hyperbolic tangent and inverse hyperbolic cotangentinverse hyperbolic sine and inverse hyperbolic cosinepower functionsingle-variable quadratic functionelementary functionlocal homeomorphismproper mapcontinuous linear operatorpseudogamma functiongeometrical continuityquasi-fibrationcontinuous group actionweak homotopy equivalencesingular functionconvex functioncontinuous real function having an interval for domainbundle mapGauss maphomotopyuniformly continuous functioncellular mapdiffeomorphismexpansive homeomorphismk times continuously differentiable functionsubmersionimmersioncontinuously differentiable functiontopological homomorphismproper group action

+15

Where this came from

wikidata · CC0 1.0

Source repository Registry entry point

Drafted structure

Bundle to layer to finding to question, as the second pass will find it: 4 bundles · 8 layers · 8 findings · 16 questions.

Definition What continuity means.

Definitions must be exact.

Epsilon-delta

Formal definition.

Epsilon-delta

Formal definition.

  1. How does the epsilon-delta definition express continuity at a point? definition
  2. How is it applied to this function? action

Test

Checking.

Test

Checking continuity.

  1. Is this function continuous on its domain? boundary
  2. Where are its discontinuities, and of what type? definition
Theorems Key results.

Continuity guarantees results.

IVT

Intermediate values.

IVT

Intermediate value theorem.

  1. How does the intermediate value theorem guarantee a root here? definition
  2. What conditions must hold? boundary

EVT

Extreme values.

EVT

Extreme value theorem.

  1. Why does a continuous function on a closed interval attain a maximum and minimum? definition
  2. What fails on open intervals? boundary
Variants Stronger notions.

Notions differ.

Uniform

Uniform continuity.

Uniform

Uniform continuity.

  1. Is the function uniformly or Lipschitz continuous? definition
  2. Why does it matter? definition

Topology

General spaces.

Topology

Topological continuity.

  1. How is continuity defined using open sets? definition
  2. What is a homeomorphism? definition
Learning Teaching.

Examples help.

Examples

Worked examples.

Examples

Examples.

  1. Which examples show continuous but not differentiable functions? definition
  2. How can they be graphed? action

Misconceptions

Common errors.

Misconceptions

Misconceptions.

  1. Which misconceptions about continuity are common? provenance
  2. How can they be corrected? action

What the second pass must settle

  • Should kinds of continuity be separate entries?
  • How should proofs be linked?
  • How should topological continuity be linked?
Draft as JSON