Math 2.0

Real functions Proof

Proof that a function with nonnegative derivative is increasing

Builds on Mean value theoremInterval

Proof

We apply the mean value theorem on each subinterval.

  1. Let x,y∈Ix, y \in I with x≤yx \le y. We must show f(x)≤f(y)f(x) \le f(y).
  2. If x=yx = y, then f(x)=f(y)f(x) = f(y). So assume x<yx < y.
  3. Since II is an interval, every zz with x≤z≤yx \le z \le y lies in II. So [x,y]⊆I[x, y] \subseteq I.
  4. Each z∈(x,y)z \in (x, y) is an interior point of II, because the open interval (x,y)(x, y) contains zz and lies in II.
  5. So ff is differentiable at each point of (x,y)(x, y), with f′(z)≥0f'(z) \ge 0 there.
  6. The function ff is continuous on II, so it is continuous on [x,y][x, y].
  7. By the mean value theorem, there is c∈(x,y)c \in (x, y) with f(y)−f(x)=f′(c)(y−x)f(y) - f(x) = f'(c)(y - x).
  8. Here f′(c)≥0f'(c) \ge 0 by step 5, and y−x>0y - x > 0. So f(y)−f(x)≥0f(y) - f(x) \ge 0.
  9. Thus f(x)≤f(y)f(x) \le f(y) whenever x≤yx \le y in II, so ff is increasing on II.

Idea

The mean value theorem writes f(y)−f(x)f(y) - f(x) as a value of f′f' times y−xy - x. A nonnegative derivative and a positive length give a nonnegative change. So ff never goes down.

Reading path

To understand this page from the ground up, read these 136 pages in order. The foundations come first. The path takes the proofs Proof of the limit of a sum of sequences, Proof of Rolle's theorem, 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 limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the linearity of differentiation, Proof of the sequential characterization of limits, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of Fermat's theorem on stationary points, Proof that limits preserve non-strict inequalities, Proof of the epsilon-delta form of the limit of a function, 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, Proof of the mean value theorem from Rolle's theorem, Proof of the constant multiple rule.

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

Where this leads

247 pages build on this one.

Further reading

  1. Web https://proofwiki.org/wiki/Real_Function_with_Positive_Derivative_is_Increasing
About this page
Tag931VMFD34H
URIm2://931VMFD34H
Kindproof
WritingW0 stub
MSC26A24
ProvesA function with nonnegative derivative is increasing
UsesMean value theorem, Interval
Changed4 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.