Math 2.0

Real functions Proof

Proof that sums and products of continuous functions are continuous

Builds on Continuous functionContinuity is equivalent to sequential continuitySequential continuityLimit of a sum of sequencesLimit of a product of sequences

Proof

We use sequential continuity and the limit rules for sequences.

  1. Let D⊆ℝD \subseteq \R and let f,g ⁣:D→ℝf, g \colon D \to \R be continuous.
  2. By the equivalence of continuity and sequential continuity, ff and gg are sequentially continuous.
  3. Let (xn)(x_n) be a sequence in DD that converges to a point a∈Da \in D. By step 2, f(xn)→f(a)f(x_n) \to f(a) and g(xn)→g(a)g(x_n) \to g(a).
  4. By the sum rule for limits, (f+g)(xn)=f(xn)+g(xn)→f(a)+g(a)=(f+g)(a)(f + g)(x_n) = f(x_n) + g(x_n) \to f(a) + g(a) = (f + g)(a).
  5. By the product rule for limits, (fg)(xn)=f(xn)g(xn)→f(a)g(a)=(fg)(a)(f g)(x_n) = f(x_n) g(x_n) \to f(a) g(a) = (f g)(a).
  6. Since (xn)(x_n) and aa were arbitrary, f+gf + g and fgf g are sequentially continuous.
  7. By the equivalence again, f+gf + g and fgf g are continuous.

Idea

Sequential continuity turns the question into one about limits of sequences. The sum and product rules for limits of sequences then give the result at once.

Reading path

To understand this page from the ground up, read these 67 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof of the epsilon-delta form of continuity, Proof that there is a natural number n with 1/n below any positive epsilon, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof that continuity is equivalent to sequential continuity.

  1. Construction of the reals from Cauchy sequences concept
  2. Element definition
  3. Integers object
  4. Limit point of a set definition
  5. Ordered field definition
  6. Proposition concept
  7. Rational numbers object
  8. Real numbers object
  9. Sequence definition
  10. Set bounded above definition
  11. Set definition
  12. Ring of integers statement
  13. Limit of a function definition
  14. Archimedean ordered field definition
  15. Implication definition
  16. Predicate definition
  17. The reals form a field statement
  18. Real sequence definition
  19. Supremum definition
  20. Ordered pair definition
  21. Continuity at a point definition
  22. The rationals form a field statement
  23. Universal quantifier definition
  24. Bounded sequence definition
  25. Limit of a sequence definition
  26. Completeness axiom statement
  27. Cartesian product definition
  28. Continuous function definition
  29. Subset definition
  30. The rationals form an ordered field statement
  31. Convergent sequence definition
  32. Sequential continuity definition
  33. Binary relation definition
  34. Order on the reals definition
  35. Antisymmetric relation definition
  36. Reflexive relation definition
  37. Transitive relation definition
  38. Interval definition
  39. Preorder definition
  40. Proof of the epsilon-delta form of continuity proof
  41. Partial order definition
  42. Epsilon-delta definition of continuity statement
  43. Total order definition
  44. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  45. The reals form an ordered field statement
  46. Absolute value definition
  47. Proof of the approximation property of the supremum proof
  48. Proof of the triangle inequality for real numbers proof
  49. Proof that the absolute value is multiplicative proof
  50. Proof of the epsilon-N definition of the limit of a sequence proof
  51. Approximation property of the supremum statement
  52. Triangle inequality statement
  53. Absolute value is multiplicative statement
  54. Epsilon-N definition of the limit of a sequence statement
  55. Proof that the reals are Archimedean from the completeness axiom proof
  56. Proof of the limit of a sum of sequences proof
  57. Proof that convergent sequences are bounded proof
  58. The reals are Archimedean statement
  59. Proof that there is a natural number n with 1/n below any positive epsilon proof
  60. Limit of a sum of sequences statement
  61. Convergent sequences are bounded statement
  62. There is a natural number n with 1/n below any positive epsilon statement
  63. Proof of the limit of a product of sequences proof
  64. Proof that continuity is equivalent to sequential continuity proof
  65. Limit of a product of sequences statement
  66. Continuity is equivalent to sequential continuity statement
  67. Proof that sums and products of continuous functions are continuous proof

Where this leads

329 pages build on this one.

Further reading

  1. Web https://www.jirka.org/ra/, Proposition 3.2.5
About this page
TagYSQX3J8KE7
URIm2://YSQX3J8KE7
Kindproof
WritingW0 stub
MSC26A15
ProvesProduct of continuous functions is continuous
UsesContinuous function, Continuity is equivalent to sequential continuity, Sequential continuity, Limit of a sum of sequences, Limit of a product of sequences
Changed4 November 2026

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