Math 2.0

Real functions Counterexample

A continuous nowhere differentiable function

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

Counterexample

Let φ ⁣:ℝ→ℝ\varphi \colon \mathbb{R} \to \mathbb{R} be the zigzag function with φ(x)=∣x∣\varphi(x) = |x| for −1≤x≤1-1 \le x \le 1 and φ(x+2)=φ(x)\varphi(x + 2) = \varphi(x) for all xx. Define f(x)=∑n=0∞(34)nφ(4nx).f(x) = \sum_{n=0}^{\infty} \Big( \frac{3}{4} \Big)^n \varphi(4^n x). The function ff is continuous on ℝ\mathbb{R}, but it is not differentiable at any point.

Remarks

Since 0≤φ≤10 \le \varphi \le 1, the M-test with Mn=(3/4)nM_n = (3/4)^n and the continuity of uniform sums give continuity. Fix xx. For each mm, choose δm=±124−m\delta_m = \pm \frac{1}{2} 4^{-m} so that no integer lies strictly between 4mx4^m x and 4m(x+δm)4^m (x + \delta_m). Then ∣f(x+δm)−f(x)δm∣≥3m+12\Big| \frac{f(x + \delta_m) - f(x)}{\delta_m} \Big| \ge \frac{3^m + 1}{2}. This tends to infinity while δm→0\delta_m \to 0, so f′(x)f'(x) does not exist. This function replaces the cosine in the Weierstrass function by a piecewise linear zigzag.

Reading path

To understand this page from the ground up, read these 32 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 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…statementA continuous nowhere differentiable function, counterexample, no evidence levelA continuous nowhere …counterexample
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. A continuous nowhere differentiable function counterexample

Further reading

  1. Web https://en.wikipedia.org/wiki/Weierstrass_function_(nowhere-differentiable_function), Construction
About this page
Tag0RXK2Z808Q
URIm2://0RXK2Z808Q
Kindcounterexample
WritingW0 stub
MSC26A27
UsesWeierstrass M-test, Continuity of the sum of a uniformly convergent series of functions, Differentiable function

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)
  • 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.