Math 2.0

statement proposition

Polynomial functions are 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.

Explanations

L1 human

Polynomials are built from xx and constants by addition and multiplication. These operations keep continuity, so each polynomial is continuous.

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To read this node, read these 17 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
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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. Polynomial functions are continuous statement

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3 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Polynomial, Polynomial functions

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