Math 2.0

Real functions Proof

Proof that the continuous image of a closed interval is a closed interval

Builds on Extreme value theoremIntermediate value theoremContinuous functionContinuity at a point

Proof

We combine the extreme value theorem with the intermediate value theorem.

  1. Let a≤ba \le b and let f ⁣:[a,b]→ℝf \colon [a, b] \to \R be continuous.
  2. By the extreme value theorem, there are c,d∈[a,b]c, d \in [a, b] with f(c)≤f(x)≤f(d)f(c) \le f(x) \le f(d) for all x∈[a,b]x \in [a, b].
  3. Let m=f(c)m = f(c) and M=f(d)M = f(d). These are the minimum and the maximum of ff on [a,b][a, b], and step 2 gives f([a,b])⊆[m,M]f([a, b]) \subseteq [m, M].
  4. Conversely, let y∈[m,M]y \in [m, M]. If c=dc = d, then m=Mm = M, so y=f(c)∈f([a,b])y = f(c) \in f([a, b]).
  5. Otherwise, let p=min⁡(c,d)p = \min(c, d) and q=max⁡(c,d)q = \max(c, d). Then a≤p<q≤ba \le p < q \le b.
  6. Let hh be the restriction of ff to [p,q][p, q]. Let z∈[p,q]z \in [p, q] and let VV be an open interval containing h(z)h(z).
  7. Continuity of ff at zz gives an open interval UU containing zz with f(x)∈Vf(x) \in V for all x∈U∩[a,b]x \in U \cap [a, b]. Since U∩[p,q]⊆U∩[a,b]U \cap [p, q] \subseteq U \cap [a, b], hh is continuous at zz.
  8. So hh is continuous. The numbers h(p)h(p) and h(q)h(q) are mm and MM in some order, so yy lies between h(p)h(p) and h(q)h(q).
  9. By the intermediate value theorem, there is x∈[p,q]x \in [p, q] with f(x)=h(x)=yf(x) = h(x) = y. So y∈f([a,b])y \in f([a, b]).
  10. By steps 3, 4 and 9, f([a,b])=[m,M]f([a, b]) = [m, M].

Idea

The extreme value theorem gives the smallest and largest values mm and MM, both attained. The intermediate value theorem, used on the interval between a minimum point and a maximum point, fills in every value between them. Nothing outside [m,M][m, M] can be hit, by the choice of mm and MM.

Reading path

To understand this page from the ground up, read these 116 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 that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, 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 double negation law by a truth table, Proof of the extreme value theorem by the Bolzano–Weierstrass theorem, Proof of the uniqueness of limits, Proof of strong induction from ordinary induction, Proof of the squeeze theorem for sequences, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof that limits preserve non-strict inequalities, Proof of the intermediate value theorem by the supremum, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof that continuity is equivalent to sequential continuity, Proof of the boundedness theorem by Bolzano–Weierstrass.

  1. Construction of the reals from Cauchy sequences concept
  2. Definition by recursion concept
  3. Element definition
  4. Integers object
  5. Limit point of a set definition
  6. Mathematical induction concept
  7. Natural numbers object
  8. Ordered field definition
  9. Peano axioms concept
  10. Proposition concept
  11. Rational numbers object
  12. Real numbers object
  13. Sequence definition
  14. Set bounded above definition
  15. Set definition
  16. Ring of integers statement
  17. Limit of a function definition
  18. Archimedean ordered field definition
  19. Successor function definition
  20. Conjunction definition
  21. Disjunction definition
  22. Implication definition
  23. Negation definition
  24. Predicate definition
  25. The reals form a field statement
  26. Real sequence definition
  27. Subsequence definition
  28. Infimum definition
  29. Supremum definition
  30. Ordered pair definition
  31. Continuity at a point definition
  32. Addition of natural numbers definition
  33. Proof of the Archimedean property in an Archimedean ordered field proof
  34. The rationals form a field statement
  35. Biconditional definition
  36. The successor function is injective statement
  37. Zero is not a successor statement
  38. Truth table concept
  39. Modus ponens concept
  40. Universal quantifier definition
  41. Bounded sequence definition
  42. Limit of a sequence definition
  43. Monotone sequence definition
  44. Completeness axiom statement
  45. Cartesian product definition
  46. Continuous function definition
  47. Archimedean property statement
  48. Proof of the commutativity of addition by induction proof
  49. Order on the natural numbers definition
  50. Subset definition
  51. The rationals form an ordered field statement
  52. Propositional logic concept
  53. Mathematical proof concept
  54. Convergent sequence definition
  55. Sequential continuity definition
  56. Binary relation definition
  57. Order on the reals definition
  58. Commutativity of addition statement
  59. Proof of strong induction from ordinary induction proof
  60. Tautology definition
  61. Antisymmetric relation definition
  62. Reflexive relation definition
  63. Transitive relation definition
  64. Interval definition
  65. Strong induction statement
  66. Logical equivalence definition
  67. Preorder definition
  68. Proof of the epsilon-delta form of continuity proof
  69. Proof of the well-ordering principle from strong induction proof
  70. Proof of the double negation law by a truth table proof
  71. Partial order definition
  72. Epsilon-delta definition of continuity statement
  73. Well-ordering principle statement
  74. Double negation law statement
  75. Total order definition
  76. Proof by contradiction concept
  77. Proof that every sequence has a monotone subsequence, by peaks proof
  78. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  79. Every sequence has a monotone subsequence statement
  80. The reals form an ordered field statement
  81. Absolute value definition
  82. Proof of the approximation property of the supremum proof
  83. Proof of the triangle inequality for real numbers proof
  84. Proof of the epsilon-N definition of the limit of a sequence proof
  85. Approximation property of the supremum statement
  86. Triangle inequality statement
  87. Epsilon-N definition of the limit of a sequence statement
  88. Proof of the intermediate value theorem by the supremum proof
  89. Proof that the reals are Archimedean from the completeness axiom proof
  90. Proof of the limit of a sum of sequences proof
  91. Proof of the squeeze theorem for sequences proof
  92. Proof of the uniqueness of limits proof
  93. Proof that limits preserve non-strict inequalities proof
  94. Intermediate value theorem statement
  95. The reals are Archimedean statement
  96. Proof that convergent sequences are bounded proof
  97. Proof of the monotone convergence theorem for sequences proof
  98. Proof that there is a natural number n with 1/n below any positive epsilon proof
  99. Limit of a sum of sequences statement
  100. Uniqueness of limits statement
  101. Limits preserve non-strict inequalities statement
  102. Squeeze theorem statement
  103. Convergent sequences are bounded statement
  104. Monotone convergence theorem for sequences statement
  105. There is a natural number n with 1/n below any positive epsilon statement
  106. Proof that a decreasing sequence bounded below converges proof
  107. Decreasing sequence bounded below converges statement
  108. Proof that continuity is equivalent to sequential continuity proof
  109. Continuity is equivalent to sequential continuity statement
  110. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  111. Bolzano–Weierstrass theorem statement
  112. Proof of the boundedness theorem by Bolzano–Weierstrass proof
  113. Boundedness theorem statement
  114. Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
  115. Extreme value theorem statement
  116. Proof that the continuous image of a closed interval is a closed interval proof

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Intermediate_value_theorem, Theorem
About this page
Tag2TR8XV1ZBW
URIm2://2TR8XV1ZBW
Kindproof
WritingW0 stub
MSC26A15
ProvesContinuous image of a closed interval is a closed interval
UsesExtreme value theorem, Intermediate value theorem, Continuous function, Continuity at a point
Changed4 November 2026

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