Math 2.0

statement proposition

Product of continuous functions is continuous

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.

Explanations

L1 human

If f(x)f(x) and g(x)g(x) change only a little when xx changes a little, so does f(x)g(x)f(x) g(x).

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To read this node, read these 16 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…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

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

Sources

  1. Web https://en.wikipedia.org/wiki/Continuous_function, Rules for continuity

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