Math 2.0

Real functions Theorem

Continuity is equivalent to sequential continuity

Builds on Continuous functionSequential continuityEpsilon-delta definition of continuity

Statement

Let D⊆ℝD \subseteq \mathbb{R} and let f ⁣:D→ℝf \colon D \to \mathbb{R} be a function. Then ff is continuous if and only if ff is sequentially continuous.

Remarks

The forward direction follows from the epsilon-delta form of continuity. For the converse, suppose that ff is not continuous at aa. Then there is ε>0\varepsilon > 0 and for each nn a point xnx_n with ∣xn−a∣<1/n|x_n - a| < 1/n and ∣f(xn)−f(a)∣≥ε|f(x_n) - f(a)| \ge \varepsilon. The equivalence holds in all metric spaces. Mathlib states it for sequential spaces, as continuous_iff_seqContinuous.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionEpsilon-delta definition of continuity, statement, no evidence levelEpsilon-delta definit…statementSequential continuity, definition, no evidence levelSequential continuitydefinitionContinuity is equivalent to sequential continuity, 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. Limit point of a set definition
  2. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Continuity at a point definition
  6. Limit of a sequence definition
  7. Continuous function definition
  8. Epsilon-delta definition of continuity statement
  9. Sequential continuity definition
  10. Continuity is equivalent to sequential continuity statement

Where this leads

16 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Definition in terms of limits of sequences
About this page
Tag0PAQM3H57X
URIm2://0PAQM3H57X
Kindstatement, theorem
WritingW0 stub
MSC26A15
StatusThis copy of the site has no build data. Run m2 status 0PAQM3H57X.
UsesContinuous function, Sequential continuity, Epsilon-delta definition of continuity

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