Math 2.0

General topology Definition

Equicontinuity

Builds on Metric spaceEpsilon-delta continuity in metric spaces

Definition

Let (X,dX)(X, d_X) and (Y,dY)(Y, d_Y) be metric spaces and let ℱ\mathcal{F} be a set of functions from XX to YY. The family ℱ\mathcal{F} is equicontinuous at a point a∈Xa \in X when the property that follows holds. For each real ε>0\varepsilon > 0 there is a real δ>0\delta > 0 such that dY(f(x),f(a))<εd_Y(f(x), f(a)) < \varepsilon for all f∈ℱf \in \mathcal{F} and all x∈Xx \in X with dX(x,a)<δd_X(x, a) < \delta. The family ℱ\mathcal{F} is equicontinuous when it is equicontinuous at each point of XX.

Remarks

The number δ\delta depends on ε\varepsilon and aa, but not on ff. Each finite set of continuous functions is equicontinuous. The family of all functions f ⁣:ℝ→ℝf \colon \mathbb{R} \to \mathbb{R} that are Lipschitz with constant 11 is equicontinuous: take δ=ε\delta = \varepsilon. The family fn(x)=xnf_n(x) = x^n on [0,1][0, 1] is not equicontinuous at x=1x = 1.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

Show the path as a graph
Element, definition, no evidence levelElementdefinitionProposition, concept, no evidence levelPropositionconceptReal numbers, object, no evidence levelReal numbersobjectSet, definition, no evidence levelSetdefinitionConjunction, definition, no evidence levelConjunctiondefinitionDisjunction, definition, no evidence levelDisjunctiondefinitionImplication, definition, no evidence levelImplicationdefinitionPredicate, definition, no evidence levelPredicatedefinitionEmpty set, object, no evidence levelEmpty setobjectFamily of sets, definition, no evidence levelFamily of setsdefinitionOrdered pair, definition, no evidence levelOrdered pairdefinitionIntersection, definition, no evidence levelIntersectiondefinitionUnion, definition, no evidence levelUniondefinitionExistential quantifier, definition, no evidence levelExistential quantifierdefinitionUniversal quantifier, definition, no evidence levelUniversal quantifierdefinitionCartesian product, definition, no evidence levelCartesian productdefinitionSubset, definition, no evidence levelSubsetdefinitionUnion of a family of sets, definition, no evidence levelUnion of a family of …definitionUniqueness quantifier, definition, no evidence levelUniqueness quantifierdefinitionBinary relation, definition, no evidence levelBinary relationdefinitionTopological space, definition, no evidence levelTopological spacedefinitionFunction, definition, no evidence levelFunctiondefinitionPreimage of a set, definition, no evidence levelPreimage of a setdefinitionMetric space, definition, no evidence levelMetric spacedefinitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionOpen sets of a metric space, statement, no evidence levelOpen sets of a metric…statementOpen sets characterised by neighbourhoods, statement, no evidence levelOpen sets characteris…statementContinuity at a point of a topological space, definition, no evidence levelContinuity at a point…definitionEpsilon-delta continuity in metric spaces, statement, no evidence levelEpsilon-delta continu…statementEquicontinuity, definition, no evidence levelEquicontinuitydefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Element definition
  2. Proposition concept
  3. Real numbers object
  4. Set definition
  5. Conjunction definition
  6. Disjunction definition
  7. Implication definition
  8. Predicate definition
  9. Empty set object
  10. Family of sets definition
  11. Ordered pair definition
  12. Intersection definition
  13. Union definition
  14. Existential quantifier definition
  15. Universal quantifier definition
  16. Cartesian product definition
  17. Subset definition
  18. Union of a family of sets definition
  19. Uniqueness quantifier definition
  20. Binary relation definition
  21. Topological space definition
  22. Function definition
  23. Preimage of a set definition
  24. Metric space definition
  25. Open ball definition
  26. Metric topology definition
  27. Open set definition
  28. Open sets of a metric space statement
  29. Open sets characterised by neighbourhoods statement
  30. Continuity at a point of a topological space definition
  31. Epsilon-delta continuity in metric spaces statement
  32. Equicontinuity definition

Where this leads

15 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Equicontinuity, Equicontinuity between metric spaces
About this page
Tag0MW2WXTHMC
URIm2://0MW2WXTHMC
Kinddefinition
WritingW1 stated
MSC54C35
DefinesEquicontinuity
UsesMetric space, Epsilon-delta continuity in metric spaces

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by