Math 2.0

General topology Theorem

Continuity is equivalent to sequential continuity in metric spaces

Builds on Continuous map of topological spacesConvergence of a sequence in a metric spaceEpsilon-delta continuity in metric spaces

Statement

Let XX and YY be metric spaces and let f ⁣:X→Yf \colon X \to Y be a function. Then ff is continuous if and only if f(xn)→f(x)f(x_n) \to f(x) for each sequence (xn)(x_n) in XX that converges to a point xx.

Remarks

The condition on sequences is sequential continuity. Proof of the converse: if ff is not continuous at xx, then by epsilon-delta continuity there are ε>0\varepsilon > 0 and points xnx_n with d(xn,x)<1/nd(x_n, x) < 1/n and d(f(xn),f(x))≥εd(f(x_n), f(x)) \ge \varepsilon. Then xn→xx_n \to x, but f(xn)↛f(x)f(x_n) \not\to f(x). The same equivalence holds at each single point. In general topological spaces, continuity implies sequential continuity, but the converse can fail.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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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 spacedefinitionConvergence of a sequence in a metric space, definition, no evidence levelConvergence of a sequ…definitionOpen ball, definition, no evidence levelOpen balldefinitionMetric topology, definition, no evidence levelMetric topologydefinitionOpen set, definition, no evidence levelOpen setdefinitionContinuous map of topological spaces, definition, no evidence levelContinuous map of top…definitionOpen 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…statementContinuity is equivalent to sequential continuity in metric spaces, statement, no evidence levelContinuity is equival…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.
  • 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. Convergence of a sequence in a metric space definition
  26. Open ball definition
  27. Metric topology definition
  28. Open set definition
  29. Continuous map of topological spaces definition
  30. Open sets of a metric space statement
  31. Open sets characterised by neighbourhoods statement
  32. Continuity at a point of a topological space definition
  33. Epsilon-delta continuity in metric spaces statement
  34. Continuity is equivalent to sequential continuity in metric spaces statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Sequences and nets
About this page
Tag0VGPKZ8MRK
URIm2://0VGPKZ8MRK
Kindstatement, theorem
WritingW0 stub
MSC54E35
StatusThis copy of the site has no build data. Run m2 status 0VGPKZ8MRK.
UsesContinuous map of topological spaces, Convergence of a sequence in a metric space, Epsilon-delta continuity in metric spaces

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