Math 2.0

Real functions Definition

Rational function

Builds on Quotient of continuous functions is continuousPolynomial functions are continuous

Definition

Let pp and qq be real polynomials, with qq not the zero polynomial. Let D={x∈ℝ:q(x)≠0}D = \{ x \in \mathbb{R} : q(x) \ne 0 \}. The function f ⁣:D→ℝf \colon D \to \mathbb{R} given by f(x)=p(x)/q(x)f(x) = p(x)/q(x) is a rational function.

Remarks

A nonzero polynomial has only finitely many roots, so DD is ℝ\mathbb{R} minus a finite set. Each rational function is continuous on DD, since polynomials are continuous and quotients of continuous functions are continuous. For example, 1/x1/x and (x2+1)/(x−2)(x^2 + 1)/(x - 2) are rational functions.

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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…statementLimit of a quotient of sequences, statement, no evidence levelLimit of a quotient o…statementPolynomial functions are continuous, statement, no evidence levelPolynomial functions …statementQuotient of continuous functions is continuous, statement, no evidence levelQuotient of continuou…statementRational function, definition, no evidence levelRational functiondefinition
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. 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. Limit of a quotient of sequences statement
  18. Polynomial functions are continuous statement
  19. Quotient of continuous functions is continuous statement
  20. Rational function definition

Where this leads

11 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Rational_function, Definitions
About this page
Tag0WCZH4AFMT
URIm2://0WCZH4AFMT
Kinddefinition
WritingW1 stated
MSC26A09
DefinesRational function
UsesQuotient of continuous functions is continuous, Polynomial functions are continuous

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