Math 2.0

Real functions Proposition

Product of continuous functions is continuous

Builds on Continuous functionLimit of a product of sequencesContinuity is equivalent to sequential continuity

Statement

Let D⊆ℝD \subseteq \mathbb{R} and let f,g ⁣:D→ℝf, g \colon D \to \mathbb{R} be continuous functions. Then the sum f+gf + g and the product fgf g, given by (fg)(x)=f(x)g(x)(f g)(x) = f(x) g(x), are continuous.

Remarks

The proof uses sequential continuity and the product rule for limits of sequences. In particular, cfc f is continuous for each constant cc. Mathlib states it for topological monoids, as Continuous.mul.

Reading path

To understand this page from the ground up, read these 16 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 pointdefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit 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…statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionEpsilon-N definition of the limit of a sequence, statement, no evidence levelEpsilon-N definition …statementSequential continuity, definition, no evidence levelSequential continuitydefinitionContinuity is equivalent to sequential continuity, statement, no evidence levelContinuity is equival…statementConvergent sequences are bounded, statement, no evidence levelConvergent sequences …statementLimit of a product of sequences, statement, no evidence levelLimit of a product of…statementProduct of continuous functions is continuous, statement, no evidence levelProduct of continuous…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. Bounded sequence definition
  7. Limit of a sequence definition
  8. Continuous function definition
  9. Epsilon-delta definition of continuity statement
  10. Convergent sequence definition
  11. Epsilon-N definition of the limit of a sequence statement
  12. Sequential continuity definition
  13. Continuity is equivalent to sequential continuity statement
  14. Convergent sequences are bounded statement
  15. Limit of a product of sequences statement
  16. Product of continuous functions is continuous statement

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Rules for continuity
About this page
Tag0VGMHCF4SC
URIm2://0VGMHCF4SC
Kindstatement, proposition
WritingW0 stub
MSC26A15
StatusThis copy of the site has no build data. Run m2 status 0VGMHCF4SC.
UsesContinuous function, Limit of a product of sequences, Continuity is equivalent to sequential continuity

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