Math 2.0

Real functions Proof

Proof that the integral of a continuous function is an antiderivative

Builds on Continuous functions are Riemann integrableDefinite integralAdditivity of the integral over intervalsFirst fundamental theorem of calculusDerivativeAntiderivative

Proof

We reduce to the first fundamental theorem on a closed interval around each point.

  1. Let II be an open interval, let f ⁣:I→ℝf \colon I \to \mathbb{R} be continuous, and let a∈Ia \in I. For x∈Ix \in I, the restriction of ff to the closed interval between aa and xx is continuous. Since continuous functions are Riemann integrable, the definite integral F(x)=∫axf(t) dtF(x) = \int_a^x f(t) \, dt is defined.
  2. Fix x0∈Ix_0 \in I. Since II is open, there are p,q∈Ip, q \in I with p<min⁡(a,x0)p < \min(a, x_0) and q>max⁡(a,x0)q > \max(a, x_0). Then [p,q]⊆I[p, q] \subseteq I, and ff is continuous on [p,q][p, q].
  3. Define G(x)=∫pxf(t) dtG(x) = \int_p^x f(t) \, dt for x∈[p,q]x \in [p, q]. We claim that F(x)=G(x)−G(a)F(x) = G(x) - G(a) for all x∈(p,q)x \in (p, q).
  4. If x>ax > a, the additivity of the integral on [p,x][p, x] split at aa gives G(x)=G(a)+∫axf(t) dt=G(a)+F(x)G(x) = G(a) + \int_a^x f(t) \, dt = G(a) + F(x).
  5. If x=ax = a, then F(a)=0=G(a)−G(a)F(a) = 0 = G(a) - G(a) by the convention for the definite integral.
  6. If x<ax < a, additivity on [p,a][p, a] split at xx gives G(a)=G(x)+∫xaf(t) dtG(a) = G(x) + \int_x^a f(t) \, dt. By the sign convention, ∫xaf(t) dt=−F(x)\int_x^a f(t) \, dt = -F(x). So F(x)=G(x)−G(a)F(x) = G(x) - G(a).
  7. The function ff is continuous at x0∈(p,q)x_0 \in (p, q). By the first fundamental theorem of calculus on [p,q][p, q], GG is differentiable at x0x_0 with G′(x0)=f(x0)G'(x_0) = f(x_0).
  8. For 0<∣h∣0 < |h| small enough, x0+h∈(p,q)x_0 + h \in (p, q). Then step 3 gives F(x0+h)−F(x0)h=G(x0+h)−G(x0)h\frac{F(x_0 + h) - F(x_0)}{h} = \frac{G(x_0 + h) - G(x_0)}{h}.
  9. So the two difference quotients have the same limit as h→0h \to 0. By the definition of the derivative, FF is differentiable at x0x_0 with F′(x0)=f(x0)F'(x_0) = f(x_0).
  10. Since x0∈Ix_0 \in I was arbitrary, FF is differentiable on II with F′=fF' = f. So FF is an antiderivative of ff on II.

Idea

Differentiability is a local property. Near any point x0x_0 we can place a closed interval [p,q][p, q] inside II. There FF differs from ∫pxf\int_p^x f only by a constant, and the first fundamental theorem gives the derivative.

Reading path

To understand this page from the ground up, read these 151 pages in order. The foundations come first. The path takes the proofs Proof that Darboux and Riemann integrals agree, 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 continuous functions on a closed bounded interval are uniformly continuous, 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 that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of the mean value theorem for integrals, Proof of Riemann's criterion for integrability, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the first fundamental theorem of calculus, Proof that limits preserve non-strict inequalities, Proof of the linearity of the integral, Proof of the intermediate value theorem by the supremum, 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 of the additivity of the integral over intervals, 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 monotonicity of the integral, Proof of the boundedness theorem by Bolzano–Weierstrass, Proof that continuous functions are Riemann integrable.

  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. Partition of an interval 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. Archimedean ordered field definition
  20. Successor function definition
  21. Mesh of a partition definition
  22. Refinement of a partition definition
  23. Riemann sum definition
  24. Conjunction definition
  25. Disjunction definition
  26. Implication definition
  27. Negation definition
  28. Predicate definition
  29. The reals form a field statement
  30. Real sequence definition
  31. Subsequence definition
  32. Infimum definition
  33. Supremum definition
  34. Ordered pair definition
  35. Continuity at a point definition
  36. Derivative definition
  37. Addition of natural numbers definition
  38. Proof of the Archimedean property in an Archimedean ordered field proof
  39. Riemann integrable function definition
  40. The rationals form a field statement
  41. The successor function is injective statement
  42. Zero is not a successor statement
  43. Biconditional definition
  44. Truth table concept
  45. Modus ponens concept
  46. Universal quantifier definition
  47. Bounded sequence definition
  48. Limit of a sequence definition
  49. Monotone sequence definition
  50. Completeness axiom statement
  51. Cartesian product definition
  52. Continuous function definition
  53. Antiderivative definition
  54. Archimedean property statement
  55. Proof of the commutativity of addition by induction proof
  56. Order on the natural numbers definition
  57. Riemann integral definition
  58. Subset definition
  59. The rationals form an ordered field statement
  60. Propositional logic concept
  61. Upper and lower Darboux sums definition
  62. Mathematical proof concept
  63. Convergent sequence definition
  64. Sequential continuity definition
  65. Binary relation definition
  66. Commutativity of addition statement
  67. Order on the reals definition
  68. Proof of strong induction from ordinary induction proof
  69. Definite integral definition
  70. Tautology definition
  71. Upper and lower Darboux integrals definition
  72. Antisymmetric relation definition
  73. Reflexive relation definition
  74. Transitive relation definition
  75. Interval definition
  76. Strong induction statement
  77. Logical equivalence definition
  78. Preorder definition
  79. Proof of the epsilon-delta form of continuity proof
  80. Proof of the epsilon-delta form of the limit of a function proof
  81. Proof of the well-ordering principle from strong induction proof
  82. Proof of the double negation law by a truth table proof
  83. Partial order definition
  84. Epsilon-delta definition of continuity statement
  85. Epsilon-delta definition of a limit statement
  86. Well-ordering principle statement
  87. Double negation law statement
  88. Total order definition
  89. Uniform continuity definition
  90. Proof by contradiction concept
  91. Proof that every sequence has a monotone subsequence, by peaks proof
  92. Uniform continuity on a set definition
  93. Proof that the reals form an ordered field, with the Cauchy sequence construction proof
  94. Every sequence has a monotone subsequence statement
  95. The reals form an ordered field statement
  96. Absolute value definition
  97. Proof of the approximation property of the supremum proof
  98. Proof of the triangle inequality for real numbers proof
  99. Proof that the absolute value is multiplicative proof
  100. Proof of the epsilon-N definition of the limit of a sequence proof
  101. Approximation property of the supremum statement
  102. Triangle inequality statement
  103. Absolute value is multiplicative statement
  104. Epsilon-N definition of the limit of a sequence statement
  105. Proof of the intermediate value theorem by the supremum proof
  106. Proof that the reals are Archimedean from the completeness axiom proof
  107. Proof of the limit of a sum of sequences proof
  108. Proof of the squeeze theorem for sequences proof
  109. Proof of the uniqueness of limits proof
  110. Proof that limits preserve non-strict inequalities proof
  111. Intermediate value theorem statement
  112. Proof of the linearity of the integral proof
  113. The reals are Archimedean statement
  114. Proof that convergent sequences are bounded proof
  115. Proof of the monotone convergence theorem for sequences proof
  116. Proof that there is a natural number n with 1/n below any positive epsilon proof
  117. Limit of a sum of sequences statement
  118. Linearity of the integral statement
  119. Proof of the monotonicity of the integral proof
  120. Uniqueness of limits statement
  121. Limits preserve non-strict inequalities statement
  122. Squeeze theorem statement
  123. Proof that Darboux and Riemann integrals agree proof
  124. Convergent sequences are bounded statement
  125. Monotone convergence theorem for sequences statement
  126. There is a natural number n with 1/n below any positive epsilon statement
  127. Monotonicity of the integral statement
  128. Darboux and Riemann integrals agree statement
  129. Proof that a decreasing sequence bounded below converges proof
  130. Proof that continuity is equivalent to sequential continuity proof
  131. Decreasing sequence bounded below converges statement
  132. Proof of Riemann's criterion for integrability proof
  133. Continuity is equivalent to sequential continuity statement
  134. Riemann's criterion for integrability statement
  135. Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
  136. Proof of the additivity of the integral over intervals proof
  137. Bolzano–Weierstrass theorem statement
  138. Additivity of the integral over intervals statement
  139. Proof that continuous functions on a closed bounded interval are uniformly continuous proof
  140. Proof of the boundedness theorem by Bolzano–Weierstrass proof
  141. Continuous functions on a closed bounded interval are uniformly continuous statement
  142. Boundedness theorem statement
  143. Proof of the first fundamental theorem of calculus proof
  144. Proof of the extreme value theorem by the Bolzano–Weierstrass theorem proof
  145. Extreme value theorem statement
  146. Proof that continuous functions are Riemann integrable proof
  147. Continuous functions are Riemann integrable statement
  148. Proof of the mean value theorem for integrals proof
  149. Mean value theorem for integrals statement
  150. First fundamental theorem of calculus statement
  151. Proof that the integral of a continuous function is an antiderivative proof

Where this leads

422 pages build on this one.

Further reading

  1. Web https://www.math.ucdavis.edu/~hunter/m125b/ch1.pdf, Theorem 1.50 and Definition 1.34
About this page
TagC30ABKJ9F6
URIm2://C30ABKJ9F6
Kindproof
WritingW0 stub
MSC26A36
ProvesThe integral of a continuous function is an antiderivative
UsesContinuous functions are Riemann integrable, Definite integral, Additivity of the integral over intervals, First fundamental theorem of calculus, Derivative, Antiderivative
Changed4 November 2026

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