Math 2.0

General topology Definition

Continuity at a point of a topological space

Builds on Topological spaceOpen sets characterised by neighbourhoodsPreimage of a set

Definition

Let XX and YY be topological spaces, let f ⁣:X→Yf \colon X \to Y be a function and let a∈Xa \in X. The function ff is continuous at aa when for each neighbourhood VV of f(a)f(a), the preimage f−1(V)f^{-1}(V) is a neighbourhood of aa.

Remarks

Equivalently, for each neighbourhood VV of f(a)f(a) there is a neighbourhood UU of aa with f(U)⊆Vf(U) \subseteq V. A function is a continuous map if and only if it is continuous at each point of XX. For metric spaces, continuity at aa is the same as the epsilon-delta condition at a.

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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 characterised by neighbourhoods, statement, no evidence levelOpen sets characteris…statementContinuity at a point of a topological space, definition, no evidence levelContinuity at a point…definition
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 characterised by neighbourhoods statement
  29. Continuity at a point of a topological space definition

Where this leads

19 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Continuity at a point
About this page
Tag07N7W5C29X
URIm2://07N7W5C29X
Kinddefinition
WritingW0 stub
MSC54C05
DefinesContinuity at a point
UsesTopological space, Open sets characterised by neighbourhoods, Preimage of a set

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