Math 2.0

statement proposition

Quotient 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 with g(x)≠0g(x) \ne 0 for all x∈Dx \in D. Then the quotient f/gf/g, given by (f/g)(x)=f(x)/g(x)(f/g)(x) = f(x)/g(x), is continuous.

Remarks

The proof uses sequential continuity and the quotient rule for limits of sequences. If gg has zeros, then f/gf/g is continuous on the set where gg is not zero. Mathlib states it as Continuous.div.

Explanations

L1 human

Division by a continuous function causes no problem away from its zeros. Near a point where gg is not zero, gg stays away from zero.

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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…statementLimit of a quotient of sequences, statement, no evidence levelLimit of a quotient o…statementQuotient of continuous functions is continuous, statement, no evidence levelQuotient of continuou…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. Limit of a quotient of sequences statement
  17. Quotient of continuous functions is continuous statement

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2 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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