Math 2.0

Real functions Limits and continuity Counterexample

Weierstrass function

Builds on Weierstrass M-testContinuity of the sum of a uniformly convergent series of functionsDifferentiable function

Counterexample

Let 0<a<10 < a < 1, and let bb be a positive odd integer with ab>1+32πa b > 1 + \frac{3}{2} \pi. The Weierstrass function is W(x)=∑n=0∞ancos⁡(bnπx).W(x) = \sum_{n=0}^{\infty} a^n \cos(b^n \pi x). The function WW is continuous on ℝ\mathbb{R}, but it is not differentiable at any point. So a continuous function need not have a derivative anywhere.

Remarks

The terms satisfy ∣ancos⁡(bnπx)∣≤an|a^n \cos(b^n \pi x)| \le a^n. The M-test and the continuity of uniform sums give continuity. G. H. Hardy later showed that the conclusion holds under the weaker condition ab≥1a b \ge 1. Karl Weierstrass presented the function to the Prussian Academy of Sciences on 18 July 1872. Bernard Bolzano had constructed such a function around 1831, but it was not published until 1922.

Reading path

To understand this page from the ground up, read these 32 pages in order. The foundations come first.

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Limit point of a set, definition, no evidence levelLimit point of a setdefinitionSequence, definition, no evidence levelSequencedefinitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionReal sequence, definition, no evidence levelReal sequencedefinitionSubsequence, definition, no evidence levelSubsequencedefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionDerivative, definition, no evidence levelDerivativedefinitionBounded sequence, definition, no evidence levelBounded sequencedefinitionCauchy sequence, definition, no evidence levelCauchy sequencedefinitionLimit of a sequence, definition, no evidence levelLimit of a sequencedefinitionMonotone sequence, definition, no evidence levelMonotone sequencedefinitionSeries, definition, no evidence levelSeriesdefinitionDifferentiable function, definition, no evidence levelDifferentiable functi…definitionCauchy sequences are bounded, statement, no evidence levelCauchy sequences are …statementEvery sequence has a monotone subsequence, statement, no evidence levelEvery sequence has a …statementMonotone convergence theorem for sequences, statement, no evidence levelMonotone convergence …statementConvergent sequence, definition, no evidence levelConvergent sequencedefinitionPointwise convergence, definition, no evidence levelPointwise convergencedefinitionPartial sum, definition, no evidence levelPartial sumdefinitionBolzano–Weierstrass theorem, statement, no evidence levelBolzano–Weierstrass t…statementUniform convergence, definition, no evidence levelUniform convergencedefinitionConvergent series, definition, no evidence levelConvergent seriesdefinitionUniform convergence on a set, definition, no evidence levelUniform convergence o…definitionCauchy criterion for sequences, statement, no evidence levelCauchy criterion for …statementA series of nonnegative terms converges iff its partial sums are bounded, statement, no evidence levelA series of nonnegati…statementSeries of functions, definition, no evidence levelSeries of functionsdefinitionUniform limit theorem, statement, no evidence levelUniform limit theoremstatementUniformly Cauchy sequence of functions, definition, no evidence levelUniformly Cauchy sequ…definitionComparison test, statement, no evidence levelComparison teststatementWeierstrass M-test, statement, no evidence levelWeierstrass M-teststatementContinuity of the sum of a uniformly convergent series of functions, statement, no evidence levelContinuity of the sum…statementWeierstrass function, counterexample, no evidence levelWeierstrass 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. Sequence definition
  3. Limit of a function definition
  4. Real sequence definition
  5. Subsequence definition
  6. Continuity at a point definition
  7. Derivative definition
  8. Bounded sequence definition
  9. Cauchy sequence definition
  10. Limit of a sequence definition
  11. Monotone sequence definition
  12. Series definition
  13. Differentiable function definition
  14. Cauchy sequences are bounded statement
  15. Every sequence has a monotone subsequence statement
  16. Monotone convergence theorem for sequences statement
  17. Convergent sequence definition
  18. Pointwise convergence definition
  19. Partial sum definition
  20. Bolzano–Weierstrass theorem statement
  21. Uniform convergence definition
  22. Convergent series definition
  23. Uniform convergence on a set definition
  24. Cauchy criterion for sequences statement
  25. A series of nonnegative terms converges iff its partial sums are bounded statement
  26. Series of functions definition
  27. Uniform limit theorem statement
  28. Uniformly Cauchy sequence of functions definition
  29. Comparison test statement
  30. Weierstrass M-test statement
  31. Continuity of the sum of a uniformly convergent series of functions statement
  32. Weierstrass function counterexample

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Weierstrass_function_(nowhere-differentiable_function), Construction
About this page
Tag01A0MK1730
URIm2://01A0MK1730
Kindcounterexample
WritingW0 stub
MSC26A27
UsesWeierstrass M-test, Continuity of the sum of a uniformly convergent series of functions, Differentiable function
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