Math 2.0

General topology Proposition

Epsilon-delta continuity in metric spaces

Builds on Continuity at a point of a topological spaceOpen sets of a metric spaceOpen ball

Statement

Let (X,dX)(X, d_X) and (Y,dY)(Y, d_Y) be metric spaces, let f ⁣:X→Yf \colon X \to Y be a function and let a∈Xa \in X. Then ff is continuous at aa if and only if for each real ε>0\varepsilon > 0 there is a real δ>0\delta > 0 such that dX(x,a)<δ  ⟹  dY(f(x),f(a))<εfor all x∈X.d_X(x, a) < \delta \implies d_Y\bigl(f(x), f(a)\bigr) < \varepsilon \quad \text{for all } x \in X.

Remarks

In terms of open balls, the condition says: for each ε>0\varepsilon > 0 there is δ>0\delta > 0 with f(B(a,δ))⊆B(f(a),ε)f(B(a, \delta)) \subseteq B(f(a), \varepsilon). The number δ\delta can depend on ε\varepsilon and on aa. When one δ\delta works for all aa, the map is uniformly continuous. For X=Y=ℝX = Y = \mathbb{R}, this is the epsilon-delta definition of continuity from calculus.

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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…statement
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.
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  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

Where this leads

18 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Continuous functions between metric spaces
About this page
Tag0EKGKY5TGH
URIm2://0EKGKY5TGH
Kindstatement, proposition
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 0EKGKY5TGH.
UsesContinuity at a point of a topological space, Open sets of a metric space, Open ball

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