Math 2.0

Real functions Proposition

Polynomial functions are continuous

Builds on Continuous functionProduct of continuous functions is continuous

Statement

Let a0,a1,…,ana_0, a_1, \dots, a_n be real numbers. Then the function p ⁣:ℝ→ℝp \colon \mathbb{R} \to \mathbb{R} given by p(x)=a0+a1x+⋯+anxnp(x) = a_0 + a_1 x + \cdots + a_n x^n is continuous.

Remarks

Constant functions and the identity x↦xx \mapsto x are continuous. Sums and products of continuous functions are continuous, so induction on nn gives the result. Mathlib states it as Polynomial.continuous.

Reading path

To understand this page from the ground up, read these 17 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…statementPolynomial functions are continuous, statement, no evidence levelPolynomial functions …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
  17. Polynomial functions are continuous statement

Where this leads

13 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Polynomial, Polynomial functions
About this page
Tag0C5TP6RYSV
URIm2://0C5TP6RYSV
Kindstatement, proposition
WritingW0 stub
MSC26A15
StatusThis copy of the site has no build data. Run m2 status 0C5TP6RYSV.
UsesContinuous function, Product of continuous functions is continuous

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