Math 2.0

Real functions Limits and continuity Counterexample

Volterra's function

Builds on DerivativeRiemann integrable functionLebesgue's criterion for Riemann integrabilityFundamental theorem of calculusAntiderivative

Counterexample

Let S⊆[0,1]S \subseteq [0, 1] be the Smith–Volterra–Cantor set. It is closed, contains no interval, and has positive Lebesgue measure. Let g(t)=t2sin⁡(1/t)g(t) = t^2 \sin(1/t) for t≠0t \ne 0 and g(0)=0g(0) = 0. Then gg is differentiable on ℝ\R, with g′(t)=2tsin⁡(1/t)−cos⁡(1/t)g'(t) = 2t \sin(1/t) - \cos(1/t) for t≠0t \ne 0 and g′(0)=0g'(0) = 0.

The complement [0,1]∖S[0, 1] \setminus S is a union of disjoint open intervals. On each such interval (a,b)(a, b), let t0t_0 be the largest t∈(0,(b−a)/2]t \in (0, (b - a)/2] with g′(t)=0g'(t) = 0. Define V(x)=g(x−a)V(x) = g(x - a) for a<x≤a+t0a < x \le a + t_0, V(x)=g(t0)V(x) = g(t_0) for a+t0≤x≤b−t0a + t_0 \le x \le b - t_0, and V(x)=g(b−x)V(x) = g(b - x) for b−t0≤x<bb - t_0 \le x < b. Set V(x)=0V(x) = 0 for x∈Sx \in S.

Then VV is differentiable at every point of [0,1][0, 1], and its derivative V′V' is bounded. At each point of SS, one has V′(x)=0V'(x) = 0, but V′V' is not continuous there. So V′V' is discontinuous on a set of positive measure, and by Lebesgue's criterion it is not Riemann integrable.

Remarks

The number t0t_0 exists because g′g' has zeros at points of (0,ε)(0, \varepsilon) for every ε>0\varepsilon > 0. Near an endpoint aa, the derivative V′(x)=g′(x−a)V'(x) = g'(x - a) keeps taking values close to 11 and close to −1-1. Every point of SS is a limit of such endpoints, which gives the discontinuity of V′V' on SS. The example shows that a function can have a bounded derivative that has no Riemann integral. So the fundamental theorem of calculus in the Riemann form ∫abf′=f(b)−f(a)\int_a^b f' = f(b) - f(a) needs the hypothesis that f′f' is Riemann integrable. The Lebesgue integral repairs this: a bounded derivative is Lebesgue integrable, and the formula holds.

Reading path

To understand this page from the ground up, read these 38 pages in order. The foundations come first.

Show the path as a graph
Limit point of a set, definition, no evidence levelLimit point of a setdefinitionLocal maximum, definition, no evidence levelLocal maximumdefinitionPartition of an interval, definition, no evidence levelPartition of an inter…definitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionLocal minimum, definition, no evidence levelLocal minimumdefinitionMesh of a partition, definition, no evidence levelMesh of a partitiondefinitionRiemann sum, definition, no evidence levelRiemann sumdefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionRiemann integrable function, definition, no evidence levelRiemann integrable fu…definitionBounded sequence, definition, no evidence levelBounded sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionContinuous function, definition, no evidence levelContinuous functiondefinitionAntiderivative, definition, no evidence levelAntiderivativedefinitionFermat's theorem on stationary points, statement, no evidence levelFermat's theorem on s…statementLebesgue's criterion for Riemann integrability, statement, no evidence levelLebesgue's criterion …statementRiemann integral, definition, no evidence levelRiemann integraldefinitionConvergent sequence, definition, no evidence levelConvergent sequencedefinitionMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementIntermediate value theorem, statement, no evidence levelIntermediate value th…statementDefinite integral, definition, no evidence levelDefinite integraldefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementAdditivity of the integral over intervals, statement, no evidence levelAdditivity of the int…statementBoundedness theorem, statement, no evidence levelBoundedness theoremstatementMonotonicity of the integral, statement, no evidence levelMonotonicity of the i…statementExtreme value theorem, statement, no evidence levelExtreme value theoremstatementMean value theorem for integrals, statement, no evidence levelMean value theorem fo…statementRolle's theorem, statement, no evidence levelRolle's theoremstatementFirst fundamental theorem of calculus, statement, no evidence levelFirst fundamental the…statementMean value theorem, statement, no evidence levelMean value theoremstatementSecond fundamental theorem of calculus, statement, no evidence levelSecond fundamental th…statementFundamental theorem of calculus, statement, no evidence levelFundamental theorem o…statementVolterra's function, counterexample, no evidence levelVolterra's functioncounterexample
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Limit point of a set definition
  2. Local maximum definition
  3. Partition of an interval definition
  4. Sequence definition
  5. Limit of a function definition
  6. Local minimum definition
  7. Mesh of a partition definition
  8. Riemann sum definition
  9. Real sequence definition
  10. Subsequence definition
  11. Continuity at a point definition
  12. Derivative definition
  13. Riemann integrable function definition
  14. Bounded sequence definition
  15. Limit of a sequence definition
  16. Monotone sequence definition
  17. Continuous function definition
  18. Antiderivative definition
  19. Fermat's theorem on stationary points statement
  20. Lebesgue's criterion for Riemann integrability statement
  21. Riemann integral definition
  22. Convergent sequence definition
  23. Monotone convergence theorem for sequences statement
  24. Every sequence has a monotone subsequence statement
  25. Intermediate value theorem statement
  26. Definite integral definition
  27. Bolzano–Weierstrass theorem statement
  28. Additivity of the integral over intervals statement
  29. Boundedness theorem statement
  30. Monotonicity of the integral statement
  31. Extreme value theorem statement
  32. Mean value theorem for integrals statement
  33. Rolle's theorem statement
  34. First fundamental theorem of calculus statement
  35. Mean value theorem statement
  36. Second fundamental theorem of calculus statement
  37. Fundamental theorem of calculus statement
  38. Volterra's function counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Volterra%27s_function
About this page
TagRARXGWQ85S
URIm2://RARXGWQ85S
Kindcounterexample
WritingW0 stub
MSC26A27, 26A24
UsesDerivative, Riemann integrable function, Lebesgue's criterion for Riemann integrability, Fundamental theorem of calculus, Antiderivative
Same as
Changed2 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)
  • edge refutes or not_instance_of out (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.